SAT Math Geometry and Trigonometry practice: 11 worksheets with answer keys

Lines, angles and triangles: set 1

Lines, angles and triangles · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. The angles of a triangle measure $x^\circ$, $2x^\circ$, and $60^\circ$. What is the value of $x$? (Grid-in)
2. Lines $l$ and $m$ are parallel and are intersected by transversal line $t$. If one of the acute angles formed is $45^\circ$, what is the measure of one of the obtuse angles formed?
   • A) $45^\circ$
   • B) $90^\circ$
   • C) $135^\circ$
   • D) $180^\circ$
3. Two intersecting lines form a pair of vertical angles with measures $(3x + 10)^\circ$ and $(5x - 20)^\circ$. What is the value of $x$? (Grid-in)
4. Triangles $ABC$ and $DEF$ are similar. The lengths of the sides of $\triangle ABC$ are $5$, $12$, and $13$. If the shortest side of $\triangle DEF$ is $15$, what is the length of the longest side of $\triangle DEF$?
   • A) $26$
   • B) $36$
   • C) $39$
   • D) $60$
5. The measure of an exterior angle of a triangle is $110^\circ$. If one of the remote interior angles is $40^\circ$, what is the measure of the other remote interior angle? (Grid-in)

Level 2: Medium (Standard SAT)

1. In an isosceles triangle, the vertex angle (the angle between the two equal sides) measures $50^\circ$. What is the measure of one of the base angles?
   • A) $50^\circ$
   • B) $65^\circ$
   • C) $80^\circ$
   • D) $130^\circ$
2. Three angles form a straight line. Their measures are $x^\circ$, $2x^\circ$, and $3x^\circ$. What is the measure of the largest angle?
   • A) $30^\circ$
   • B) $60^\circ$
   • C) $90^\circ$
   • D) $120^\circ$
3. Triangle $XYZ$ contains a line segment $AB$ parallel to $YZ$, with $A$ on $XY$ and $B$ on $XZ$. If $XA = 5$, $AY = 3$, and $AB = 10$, what is the length of $YZ$? (Grid-in)
4. The interior angles of a pentagon (5-sided polygon) measure $100^\circ$, $110^\circ$, $120^\circ$, $100^\circ$, and $x^\circ$. What is the value of $x$?
   • A) $90$
   • B) $100$
   • C) $110$
   • D) $120$
5. Lines $j$ and $k$ are parallel and cut by a transversal. Two consecutive interior angles measure $(4y + 5)^\circ$ and $(5y - 14)^\circ$. What is the measure, in degrees, of the larger of the two angles? (Grid-in)

Level 3: Hard (Advanced SAT)

1. Two sides of a triangle have lengths of $7$ and $10$. Which of the following could be the length of the third side?
   • A) $3$
   • B) $15$
   • C) $17$
   • D) $20$
2. A person who is $6$ feet tall casts a shadow that is $8$ feet long. At the exact same time of day, a nearby tree casts a shadow that is $40$ feet long. What is the height of the tree in feet? (Grid-in)
3. In a right triangle, the two acute angles measure $2a^\circ$ and $(3a - 10)^\circ$. What is the measure of the smaller of these two acute angles?
   • A) $20^\circ$
   • B) $40^\circ$
   • C) $50^\circ$
   • D) $90^\circ$
4. Lines $l_1$ and $l_2$ are parallel. A third line $l_3$ intersects $l_1$ at point $A$ and intersects $l_2$ at point $B$, forming acute angles of $40^\circ$ at the intersections. A fourth line $l_4$ intersects $l_1$ at point $C$ and intersects $l_2$ at point $D$, forming acute angles of $70^\circ$. If $l_3$ and $l_4$ intersect each other at point $P$ directly between the parallel lines, what is the measure of $\angle AP C$?
   • A) $30^\circ$
   • B) $70^\circ$
   • C) $110^\circ$
   • D) $140^\circ$
5. Triangle $ABC$ is similar to Triangle $DEF$. The perimeter of $\triangle ABC$ is $24$. If every side of $\triangle DEF$ is exactly $3$ times the length of the corresponding side of $\triangle ABC$, what is the perimeter of $\triangle DEF$? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

1. Triangle $ABC$ is an isosceles triangle with $AB = AC$. The line segment $BC$ is extended past $C$ to a point $D$, creating an exterior angle $\angle ACD$. If $\angle ACD = 130^\circ$, what is the measure of $\angle A$? (Grid-in)
2. Lines $p$ and $q$ are parallel. Transversal line $t$ is perpendicular to a second transversal line $m$. If line $m$ intersects line $p$ at a $30^\circ$ angle, what is the measure of the acute angle formed by the intersection of line $t$ and line $q$?
   • A) $30^\circ$
   • B) $60^\circ$
   • C) $90^\circ$
   • D) $120^\circ$
3. A triangle has side lengths of $8$, $15$, and $x$. If $x$ must be an integer, how many possible values are there for $x$? (Grid-in)
4. In $\triangle PQR$, the length of $PQ$ equals the length of $PR$. A line segment $QS$ is drawn such that it bisects $\angle PQR$. If the measure of $\angle P$ is $36^\circ$, what is the measure of $\angle PQS$?
   • A) $18^\circ$
   • B) $36^\circ$
   • C) $72^\circ$
   • D) $144^\circ$
5. Two parallel lines are intersected by a transversal. The bisectors of two consecutive interior angles are drawn, and they intersect at a point $P$ between the parallel lines. What is the measure of the angle formed at $P$ by these two bisectors? (Grid-in)

Lines, angles and triangles: set 2

Lines, angles and triangles · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. In a triangle, the measures of two angles are $45^\circ$ and $65^\circ$. What is the measure of the third angle? (Grid-in)
2. Two lines intersect at a point, forming four angles. If one of the angles measures $105^\circ$, what is the measure of the angle vertically opposite to it?
   • A) $75^\circ$
   • B) $85^\circ$
   • C) $105^\circ$
   • D) $180^\circ$
3. Triangle $DEF$ is similar to triangle $GHI$, where $D$ corresponds to $G$ and $E$ corresponds to $H$. If the measure of angle $D$ is $40^\circ$ and the measure of angle $H$ is $60^\circ$, what is the measure of angle $F$?
   • A) $40^\circ$
   • B) $60^\circ$
   • C) $80^\circ$
   • D) $100^\circ$
4. Lines $a$ and $b$ are parallel and are intersected by a transversal line $c$. If one of the corresponding angles measures $132^\circ$, what is the measure of the other corresponding angle? (Grid-in)
5. In right triangle $ABC$, angle $C$ is a right angle. If the measure of angle $A$ is $27^\circ$, what is the measure of angle $B$? (Grid-in)

Level 2: Medium (Standard SAT)

1. Lines $l$ and $m$ are parallel. A transversal intersects them, creating a pair of same-side interior angles. One angle measures $(3x - 10)^\circ$ and the other measures $(2x + 40)^\circ$. What is the value of $x$?
   • A) $30$
   • B) $50$
   • C) $75$
   • D) $150$
2. In triangle $XYZ$, the length of side $XY$ equals the length of side $XZ$. If the measure of angle $Y$ is $55^\circ$, what is the measure of angle $X$? (Grid-in)
3. The side lengths of a right triangle are $6$, $8$, and $10$. A second triangle is similar to the first, and its longest side has a length of $25$. What is the perimeter of the second triangle?
   • A) $24$
   • B) $50$
   • C) $60$
   • D) $150$
4. In the $xy$-plane, an angle has a measure of $x^\circ$ and another angle has a measure of $y^\circ$. The two angles are supplementary. If $x = 4y - 20$, what is the value of $y$? (Grid-in)
5. The measure of an exterior angle of a triangle is $115^\circ$. The two opposite interior angles measure $4x^\circ$ and $(2x + 25)^\circ$. What is the value of $x$?
   • A) $15$
   • B) $18$
   • C) $22.5$
   • D) $45$

Level 3: Hard (Advanced SAT)

1. The measures of the interior angles of a triangle are in the ratio $2 : 3 : 4$. What is the measure of the largest angle in the triangle? (Grid-in)
2. In triangle $PQR$, line segment $ST$ is drawn parallel to side $PR$, with $S$ on side $PQ$ and $T$ on side $QR$. If $PQ = 24$, $SQ = 8$, and $PR = 36$, what is the length of line segment $ST$?
   • A) $9$
   • B) $12$
   • C) $16$
   • D) $24$
3. Two intersecting lines form four angles. The measure of one angle is $(5x + 12)^\circ$ and the measure of an adjacent angle is $(3x + 8)^\circ$. What is the measure of the smaller of the two angles?
   • A) $20^\circ$
   • B) $68^\circ$
   • C) $112^\circ$
   • D) $180^\circ$
4. Right triangles $ABC$ and $DEF$ are similar, with $A$ corresponding to $D$. The lengths of the legs of triangle $ABC$ are $5$ and $12$. If the length of the hypotenuse of triangle $DEF$ is $39$, what is the length of the shortest leg of triangle $DEF$? (Grid-in)
5. Lines $p$ and $q$ are parallel. A transversal intersects them. The measure of an alternate exterior angle is $7x + 5$, and the corresponding alternate exterior angle on the other line is $4x + 32$. What is the value of $x$? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

1. Triangle $ABC$ is similar to triangle $XYZ$. The area of triangle $ABC$ is $45$ square inches. If the lengths of the sides of triangle $XYZ$ are $3$ times the lengths of the corresponding sides of triangle $ABC$, what is the area of triangle $XYZ$ in square inches?
   • A) $15$
   • B) $135$
   • C) $405$
   • D) $1215$
2. In triangle $MNO$, the measure of angle $M$ is $(3x + 10)^\circ$, the measure of angle $N$ is $(5x - 20)^\circ$, and the measure of angle $O$ is $(x + 10)^\circ$. Which of the following accurately describes triangle $MNO$?
   • A) It is an equilateral triangle.
   • B) It is an isosceles right triangle.
   • C) It is a scalene right triangle.
   • D) It is an isosceles acute triangle.
3. A line intersects parallel lines $l_1$ and $l_2$. A second line intersects $l_1$ and $l_2$ such that the two transversals intersect exactly on $l_1$, forming a triangle between the transversals and $l_2$. If the two angles formed on $l_1$ outside the triangle measure $110^\circ$ and $130^\circ$, what is the measure of the third angle inside the triangle resting on $l_2$? (Grid-in)
4. In right triangle $PQR$, an altitude is drawn from the right angle $Q$ to the hypotenuse $PR$, intersecting $PR$ at point $S$. If $PQ = 9$ and $QR = 12$, what is the length of the altitude $QS$? (Express as a decimal). (Grid-in)
5. In a regular polygon with $n$ sides, the measure of each interior angle is $156^\circ$. What is the value of $n$? (Grid-in)

Right triangles and trigonometry

Right triangles and trigonometry · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. In right triangle $PQR$, the right angle is at $Q$. If $\sin(P) = \frac{3}{5}$, what is the value of $\cos(R)$? (Grid-in)
2. The lengths of the two legs of a right triangle are $6$ cm and $8$ cm. What is the length of the hypotenuse, in cm?
   • A) $10$
   • B) $12$
   • C) $14$
   • D) $100$
3. In a right triangle, $\sin(30^\circ) = 0.5$. What is the value of $\cos(60^\circ)$? (Grid-in)
4. In right triangle $ABC$, angle $C$ is $90^\circ$. If $AC = 7$ and $BC = 24$, what is the length of $AB$?
   • A) $17$
   • B) $25$
   • C) $31$
   • D) $625$
5. Triangle $DEF$ is a right triangle with the right angle at $E$. If $\tan(D) = \frac{15}{8}$, what is the value of $\frac{EF}{DE}$?
   • A) $\frac{8}{15}$
   • B) $\frac{15}{17}$
   • C) $\frac{15}{8}$
   • D) $\frac{17}{8}$

Level 2: Medium (Standard SAT)

1. In a right triangle, $\cos(x^\circ) = \frac{2}{3}$. What is the value of $\sin(90^\circ - x^\circ)$? (Grid-in)
2. The diagonal of a square is $10$ units long. What is the area of the square, in square units?
   • A) $25$
   • B) $50$
   • C) $75$
   • D) $100$
3. In triangle $XYZ$, angle $Y$ is a right angle. If $\sin(X) = \frac{12}{13}$, what is the value of $\tan(Z)$?
   • A) $\frac{5}{12}$
   • B) $\frac{5}{13}$
   • C) $\frac{12}{5}$
   • D) $\frac{13}{5}$
4. An equilateral triangle has a side length of $6$ inches. What is the height of the triangle, in inches? (Hint: Use the $30^\circ-60^\circ-90^\circ$ rule).
   • A) $3$
   • B) $3\sqrt{2}$
   • C) $3\sqrt{3}$
   • D) $6\sqrt{3}$
5. A right triangle has one leg with length $x$ and a hypotenuse with length $2x$. What is the measure of the angle opposite the leg of length $x$? (Grid-in)

Level 3: Hard (Advanced SAT)

1. In a right triangle, the two acute angles are $A$ and $B$. If $\sin(A) = \frac{3}{7}$, what is the value of $\sin^2(A) + \cos^2(A)$?
   • A) $\frac{9}{49}$
   • B) $\frac{40}{49}$
   • C) $1$
   • D) It cannot be determined from the given information.
2. If $\sin(3x - 15)^\circ = \cos(2x + 10)^\circ$, what is the value of $x$? (Grid-in)
3. A 20-foot ladder is leaning against a vertical wall. The angle the ladder makes with the ground is $60^\circ$. How far is the base of the ladder from the wall, in feet? (Grid-in)
4. Triangle $ABC$ is similar to triangle $DEF$, where angle $A$ corresponds to angle $D$ and angle $C$ corresponds to angle $F$. Both $C$ and $F$ are right angles. If $\sin(A) = 0.28$, what is the value of $\sin(D)$?
   • A) $0.28$
   • B) $0.72$
   • C) $0.96$
   • D) It cannot be determined from the given information.
5. In a right triangle, the tangent of one of the acute angles is $\frac{x}{y}$. What is the tangent of the other acute angle?
   • A) $\frac{x}{y}$
   • B) $\frac{y}{x}$
   • C) $1$
   • D) $\frac{x^2}{y^2}$

Level 4: Very Hard (Slightly beyond SAT)

1. In right triangle $RST$, angle $S$ is a right angle. Point $U$ lies on the hypotenuse $RT$ such that line segment $SU$ is perpendicular to $RT$. If $\sin(R) = \frac{4}{5}$, what is the value of $\cos(\angle TSU)$?
   • A) $\frac{3}{5}$
   • B) $\frac{4}{5}$
   • C) $\frac{3}{4}$
   • D) $\frac{4}{3}$
2. A square is inscribed perfectly inside a circle with a radius of $8$ units. What is the perimeter of the square?
   • A) $32$
   • B) $32\sqrt{2}$
   • C) $64$
   • D) $64\sqrt{2}$
3. In right triangle $ABC$, the right angle is $C$. If $\sin(A) = \frac{m}{n}$, which of the following expressions represents $\tan(B)$?
   • A) $\frac{n}{m}$
   • B) $\frac{m}{\sqrt{n^2 - m^2}}$
   • C) $\frac{\sqrt{n^2 - m^2}}{m}$
   • D) $\frac{\sqrt{n^2 - m^2}}{n}$
4. The measures of two acute angles are $x^\circ$ and $y^\circ$, such that $\sin(x^\circ) = \cos(y^\circ)$. If $x = 3a - 14$ and $y = 5a + 16$, what is the value of $x$? (Grid-in)
5. Two ships leave a port at the exact same time. Ship A travels due North for $3k$ miles. Ship B travels due East for $4k$ miles. If the direct distance between the two ships is $60$ miles, what is the value of $k$? (Grid-in)

Right triangles, trigonometry and circles

Right triangles and trigonometry · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. In a right triangle $ABC$, the sine of acute angle $A$ is $\frac{7}{25}$. What is the cosine of acute angle $B$?
   • A) $\frac{7}{25}$
   • B) $\frac{24}{25}$
   • C) $\frac{25}{7}$
   • D) $\frac{7}{24}$
2. A circle in the $xy$-plane has the equation $(x + 3)^2 + (y - 5)^2 = 64$. What is the center of the circle?
   • A) $(3, -5)$
   • B) $(-3, 5)$
   • C) $(9, 25)$
   • D) $(-3, -5)$
3. What is the radius of the circle given by the equation $(x - 1)^2 + y^2 = 144$? (Grid-in)
4. In right triangle $PQR$, angle $Q$ measures $90^\circ$. If $PQ = 6$ and $QR = 8$, what is the length of the hypotenuse $PR$? (Grid-in)
5. A central angle in a circle measures $\frac{\pi}{3}$ radians. What is the measure of this angle in degrees?
   • A) $30^\circ$
   • B) $60^\circ$
   • C) $90^\circ$
   • D) $120^\circ$

Level 2: Medium (Standard SAT)

1. In a right triangle, $\sin(k^\circ) = \frac{5}{13}$. If $\cos(m^\circ) = \frac{5}{13}$ and $k$ and $m$ are acute angles, what is the value of $k + m$? (Grid-in)
2. The circle $x^2 + y^2 = 25$ is intersected by the line $y = 4$ at two points. What is the length of the straight-line segment connecting these two points of intersection?
   • A) $3$
   • B) $6$
   • C) $8$
   • D) $10$
3. A circle has a circumference of $18\pi$. What is the area of a sector of this circle with a central angle of $40^\circ$?
   • A) $2\pi$
   • B) $4\pi$
   • C) $9\pi$
   • D) $18\pi$
4. The equation of a circle is $x^2 + y^2 + 10x - 4y = 7$. What is the radius of this circle? (Grid-in)
5. In a right triangle, the tangent of one of the acute angles is $\frac{3}{4}$. What is the sine of that same angle?
   • A) $\frac{3}{7}$
   • B) $\frac{3}{5}$
   • C) $\frac{4}{5}$
   • D) $\frac{5}{3}$

Level 3: Hard (Advanced SAT)

1. The endpoints of a diameter of a circle are $(-4, 2)$ and $(6, -2)$. Which of the following is the equation of the circle?
   • A) $(x - 1)^2 + y^2 = 29$
   • B) $(x - 1)^2 + y^2 = 116$
   • C) $(x + 1)^2 + y^2 = 29$
   • D) $(x + 1)^2 + y^2 = 116$
2. An arc of a circle has a length of $5\pi$ and the radius of the circle is $12$. What is the measure of the central angle that subtends this arc, in radians? (Grid-in)
3. The equation $x^2 + y^2 - 12x + 16y = -k$ represents a circle in the $xy$-plane. If the radius of the circle is $8$, what is the value of the constant $k$? (Grid-in)
4. In $\triangle ABC$, $\angle B = 90^\circ$ and $BD$ is the altitude to the hypotenuse $AC$. If $AD = 4$ and $DC = 9$, what is the length of $BD$?
   • A) $6$
   • B) $6.5$
   • C) $13$
   • D) $36$
5. A wheel with a radius of $15$ inches rolls forward without slipping. If the wheel completes exactly $4.5$ revolutions, how many inches did the wheel travel?
   • A) $67.5\pi$
   • B) $135\pi$
   • C) $225\pi$
   • D) $1012.5\pi$

Level 4: Very Hard (Slightly beyond SAT)

1. A circle in the $xy$-plane has a center at $(3, 5)$. The line $y = -\frac{1}{2}x + 14$ is tangent to the circle at point $P$. What are the coordinates of point $P$?
   • A) $(6, 11)$
   • B) $(5, 11.5)$
   • C) $(8, 10)$
   • D) $(4, 12)$
2. A square is perfectly inscribed within a circle. If the area of the square is $100$ square units, what is the exact area of the circle?
   • A) $25\pi$
   • B) $50\pi$
   • C) $100\pi$
   • D) $200\pi$
3. If $\sin(3x - 15)^\circ = \cos(2x + 20)^\circ$, and $x$ is a positive constant, what is one possible value of $x$? (Grid-in)
4. A right triangle has side lengths of $x$, $x + 7$, and $x + 14$. What is the perimeter of the triangle? (Grid-in)
5. A circle has the equation $2x^2 + 2y^2 - 20x + 24y = -10$. What is the area of this circle?
   • A) $56\pi$
   • B) $112\pi$
   • C) $224\pi$
   • D) $448\pi$

Right-triangle trigonometry, 3D shapes and areas

Right triangles and trigonometry · 15 questions

Practice Questions

Level 1: Easy (Foundational)

1. A ladder is leaning against a vertical wall. The ladder is $10$ meters long and the base of the ladder is $6$ meters from the wall. Determine the height the ladder reaches up the wall.
   A) $4$    B) $6$    C) $8$    D) $10$
2. In a right-angled triangle, an acute angle $\theta$ has $\sin\theta = \tfrac{3}{5}$. Find $\cos\theta$.
   A) $\tfrac{3}{4}$    B) $\tfrac{4}{5}$    C) $\tfrac{5}{4}$    D) $\tfrac{5}{3}$
3. A rectangular park has a length of $12$ m and a width of $5$ m. A straight path connects opposite corners. How long is the path in meters? (Grid-in)
4. Investigate the relationship in triangle $ABC$ where $\angle C = 90^\circ$. If $\angle A = 30^\circ$ and the hypotenuse $AB=8$, what is the length of the side opposite $\angle A$?
   A) $2$    B) $4$    C) $4\sqrt{3}$    D) $8\sqrt{3}$
5. A cube has a side length of $3$. Explain the calculation to find the length of the diagonal of one of its square faces. Which is correct?
   A) $3\sqrt{2}$    B) $3\sqrt{3}$    C) $3^2+3^2$    D) $3^3$

Level 2: Medium (Standard)

1. The angle of elevation from a point on the ground to the top of a tower is $45^\circ$. If the point is $50$ m from the base, determine the height of the tower in meters.
   A) $25$    B) $50$    C) $50\sqrt{2}$    D) $100$
2. A right rectangular prism has a square base with side lengths $4$ cm and a vertical height of $7$ cm. Determine the square of the length of the space diagonal. (Grid-in)
3. In $\triangle PQR$, $\angle Q = 90^\circ$, $PQ = x$, $QR = x\sqrt{3}$. Determine the measure of $\angle P$.
   A) $30^\circ$    B) $45^\circ$    C) $60^\circ$    D) $90^\circ$
4. Show that the area of an equilateral triangle with side length $6$ can be expressed as $k\sqrt{3}$. What is $k$?
   A) $3$    B) $6$    C) $9$    D) $18$
5. A right circular cone has a base radius of $5$ and a slant height of $13$. Determine the vertical height of the cone. (Grid-in)

Level 3: Hard (Advanced)

1. Two observers are standing $100$ m apart on flat ground, looking at a hot air balloon somewhere between them in the same vertical plane. The angles of elevation are $30^\circ$ and $60^\circ$. Determine the height of the balloon in meters.
   A) $25\sqrt{3}$    B) $50\sqrt{3}$    C) $75$    D) $100$
2. In a regular hexagon $ABCDEF$ with side length $4$, determine the exact distance between parallel sides $AB$ and $DE$.
   A) $4\sqrt{3}$    B) $6\sqrt{3}$    C) $8$    D) $8\sqrt{3}$
3. A square-based right pyramid has base side length $10$ and vertical height $12$. Determine the total surface area of the four triangular faces. (Grid-in)
4. The space diagonal of a cube is $6\sqrt{3}$. Investigate the volume of this cube.
   A) $36$    B) $108$    C) $216$    D) $864$
5. In right triangle $XYZ$ with $\angle Y = 90^\circ$, $\sin X = \tfrac{2ab}{a^2+b^2}$ for positive integers $a > b$. Explain what $\tan X$ must be in terms of $a$ and $b$.
   A) $\tfrac{a^2-b^2}{2ab}$    B) $\tfrac{2ab}{a^2-b^2}$    C) $\tfrac{a^2+b^2}{a^2-b^2}$    D) $\tfrac{2ab}{a^2+b^2}$

Circles

Circles · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. A circle in the $xy$-plane has center $(0, 0)$ and radius $8$. Which of the following is an equation of the circle?
   • A) $x^2 + y^2 = 8$
   • B) $x^2 + y^2 = 16$
   • C) $x^2 + y^2 = 64$
   • D) $x^2 - y^2 = 64$
2. The equation of a circle is $(x + 5)^2 + (y - 3)^2 = 49$. What are the coordinates of the center of the circle?
   • A) $(5, 3)$
   • B) $(-5, 3)$
   • C) $(5, -3)$
   • D) $(-5, -3)$
3. A circle has a radius of $12$ cm. What is the length of an arc on the circle that corresponds to a central angle of $90^\circ$?
   • A) $3\pi$
   • B) $6\pi$
   • C) $12\pi$
   • D) $36\pi$
4. A circle has an area of $100\pi$. A sector of the circle has an area of $20\pi$. What is the measure of the central angle of the sector, in degrees? (Grid-in)
5. An angle measures $\frac{\pi}{4}$ radians. What is the measure of this angle in degrees? (Grid-in)

Level 2: Medium (Standard SAT)

1. A circle in the $xy$-plane has center $(3, 4)$ and passes through the origin $(0, 0)$. Which of the following is an equation of the circle?
   • A) $(x - 3)^2 + (y - 4)^2 = 5$
   • B) $(x - 3)^2 + (y - 4)^2 = 7$
   • C) $(x - 3)^2 + (y - 4)^2 = 25$
   • D) $(x + 3)^2 + (y + 4)^2 = 25$
2. The equation of a circle is $x^2 + y^2 + 8x - 2y = 8$. What is the radius of the circle? (Grid-in)
3. In a circle with radius $r$, an arc with length $5\pi$ is subtended by a central angle of $\frac{\pi}{2}$ radians. What is the value of $r$? (Grid-in)
4. A circle in the $xy$-plane has center $(-2, 1)$ and a radius of $13$. Which of the following points lies on the circle?
   • A) $(3, 13)$
   • B) $(10, 6)$
   • C) $(11, 1)$
   • D) $(-2, 12)$
5. The circle given by the equation $(x - 6)^2 + (y + 1)^2 = 81$ is shifted 4 units to the left and 3 units up. What is the $y$-coordinate of the center of the new circle? (Grid-in)

Level 3: Hard (Advanced SAT)

1. A circle in the $xy$-plane is defined by the equation $(x - 5)^2 + (y - 5)^2 = 36$. The line $x = 5$ intersects the circle at two points, $A$ and $B$. What is the length of line segment $AB$?
   • A) $6$
   • B) $12$
   • C) $18$
   • D) $36$
2. The equation of a circle is $x^2 - 12x + y^2 + 14y = c$, where $c$ is a constant. If the radius of the circle is $10$, what is the value of $c$? (Grid-in)
3. The center of a circle in the $xy$-plane is $(8, -3)$. The circle is tangent to the $x$-axis. Which of the following is an equation of the circle?
   • A) $(x - 8)^2 + (y + 3)^2 = 9$
   • B) $(x - 8)^2 + (y + 3)^2 = 64$
   • C) $(x + 8)^2 + (y - 3)^2 = 9$
   • D) $(x + 8)^2 + (y - 3)^2 = 64$
4. The circle represented by the equation $(x + 2)^2 + (y - 10)^2 = 144$ has a center at $(h, k)$ and a radius of $r$. What is the greatest possible $y$-coordinate for any point on the circle? (Grid-in)
5. Two identical circles each have a radius of $10$. The center of one circle lies exactly on the circumference of the other circle. What is the distance between their centers? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

1. The equation of a circle is $2x^2 + 2y^2 - 16x + 20y = 14$. What is the area of the circle?
   • A) $41\pi$
   • B) $48\pi$
   • C) $82\pi$
   • D) $96\pi$
2. A circle in the $xy$-plane has a center at $(4, 3)$. A tangent line is drawn to the circle at the point $(8, 6)$. What is the slope of the tangent line?
   • A) $-\frac{4}{3}$
   • B) $-\frac{3}{4}$
   • C) $\frac{3}{4}$
   • D) $\frac{4}{3}$
3. In the $xy$-plane, the circle defined by $(x - 2)^2 + (y - 4)^2 = r^2$ intersects the $y$-axis at exactly one point. What is the value of $r^2$? (Grid-in)
4. A sector of a circle has an area of $18\pi$ and a central angle of $\frac{3\pi}{4}$ radians. What is the length of the arc that bounds this sector? (Grid-in)
5. A circle passes through the points $(0, 0)$, $(0, 10)$, and $(24, 0)$ in the $xy$-plane. What is the diameter of the circle? (Grid-in)

Area, surface area and volume

Area and volume · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. A right rectangular prism has a volume of $120$ cubic inches. If the length is $5$ inches and the width is $6$ inches, what is the height of the prism? (Grid-in)
2. A right circular cylinder has a volume of $45\pi$. If the height of the cylinder is $5$, what is the radius?
   • A) $3$
   • B) $9$
   • C) $15$
   • D) $40$
3. The total surface area of a cube is $150$ square centimeters. What is the volume of the cube, in cubic centimeters? (Grid-in)
4. A sphere has a radius of $r$. If the radius of the sphere is tripled, by what factor does the volume of the sphere increase?
   • A) $3$
   • B) $6$
   • C) $9$
   • D) $27$
5. A right circular cone has a volume of $24\pi$ and a base radius of $6$. What is the height of the cone? (Grid-in)

Level 2: Medium (Standard SAT)

1. A rectangular prism has dimensions of $x$, $2x$, and $3x$. If the total surface area of the prism is $88$ square units, what is the value of $x$? (Grid-in)
2. Solid A and Solid B are geometrically similar. The surface area of Solid A is exactly $4$ times the surface area of Solid B. The volume of Solid A is how many times the volume of Solid B?
   • A) $2$
   • B) $4$
   • C) $8$
   • D) $16$
3. The volume of a pyramid is $V$. The base of the pyramid is a square with side length $s$. Which of the following equations correctly expresses the height $h$ of the pyramid in terms of $V$ and $s$?
   • A) $h = \frac{3V}{s^2}$
   • B) $h = \frac{V}{3s^2}$
   • C) $h = \frac{\sqrt{3V}}{s}$
   • D) $h = 3Vs^2$
4. Cube A has a volume of $64$ cubic units. Cube B has a side length that is $2$ units longer than the side length of Cube A. What is the volume of Cube B? (Grid-in)
5. A hemisphere (half of a sphere) has a volume of $18\pi$. What is the radius of the hemisphere?
   • A) $3$
   • B) $6$
   • C) $9$
   • D) $27$

Level 3: Hard (Advanced SAT)

1. The volume of a right circular cylinder is $45\pi$. If the radius is halved and the height is tripled, what is the volume of the new cylinder?
   • A) $33.75\pi$
   • B) $67.5\pi$
   • C) $135\pi$
   • D) $337.5\pi$
2. A right circular cone and a right circular cylinder have the exact same radius and the exact same volume. If the height of the cone is $12$, what is the height of the cylinder? (Grid-in)
3. A composite solid is formed by attaching a right circular cone perfectly on top of a right circular cylinder. Both the cone and the cylinder have a radius of $3$. If the cylinder has a height of $4$ and the cone also has a height of $4$, what is the total volume of the composite solid? (Grid-in)
4. A rectangular pool is $10$ meters long and $5$ meters wide. It is filled with water to a depth of $2$ meters. If water is pumped out of the pool at a constant rate of $4$ cubic meters per hour, how many hours will it take to completely empty the pool? (Grid-in)
5. The surface area of a sphere is $36\pi$ square units. What is the volume of the sphere?
   • A) $12\pi$
   • B) $27\pi$
   • C) $36\pi$
   • D) $108\pi$

Level 4: Very Hard (Slightly beyond SAT)

1. A right rectangular prism has a volume represented by the expression $x^3 - 9x$. The area of the rectangular base of the prism is represented by $x^2 + 3x$. If $x > 3$, which expression represents the height of the prism?
   • A) $x - 3$
   • B) $x + 3$
   • C) $x^2 - 3x$
   • D) $x(x - 3)$
2. A cylinder holds $100$ mL of liquid. A geometrically similar cylinder is created such that all of its 1-dimensional measurements (radius and height) are exactly $2.5$ times larger than the original cylinder's measurements. What is the capacity of the larger cylinder, in mL? (Grid-in)
3. A solid metal cone has a radius of $6$ cm and a height of $10$ cm. The cone is melted down completely and the metal is used to cast identical smaller cones, each with a radius of $2$ cm and a height of $5$ cm. How many small cones can be produced? (Grid-in)
4. A cube is perfectly inscribed inside a sphere of radius $\sqrt{3}$. What is the volume of the cube?
   • A) $3\sqrt{3}$
   • B) $8$
   • C) $12$
   • D) $24\sqrt{3}$
5. The volume of a sphere is $V$. If the radius of the sphere is decreased by $50%$, what is the surface area of the new sphere in terms of the original volume $V$?
   • A) It cannot be determined without the numerical value of the radius.
   • B) $\pi (\frac{3V}{4\pi})^{2/3}$
   • C) $(\frac{9V^2}{16\pi})^{1/3}$
   • D) $\pi (\frac{3V}{32\pi})^{2/3}$

Area and volume

Area and volume · 20 questions

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. A rectangle has a length of $12$ inches and a width of $5$ inches. What is the area of the rectangle, in square inches? (Grid-in)
2. What is the volume of a rectangular prism with a length of $4$ cm, a width of $3$ cm, and a height of $10$ cm?
   • A) $17$
   • B) $60$
   • C) $120$
   • D) $144$
3. A circle has a radius of $6$ meters. Which of the following is the area of the circle, in square meters?
   • A) $6\pi$
   • B) $12\pi$
   • C) $36\pi$
   • D) $144\pi$
4. The area of a triangle is $40$ square centimeters. If the base of the triangle is $10$ centimeters, what is the height of the triangle in centimeters? (Grid-in)
5. A right circular cylinder has a radius of $3$ and a height of $5$. What is its volume?
   • A) $15\pi$
   • B) $30\pi$
   • C) $45\pi$
   • D) $75\pi$

Level 2: Medium (Standard SAT)

1. A square has an area of $81$ square inches. What is the perimeter of the square, in inches?
   • A) $9$
   • B) $18$
   • C) $36$
   • D) $81$
2. The volume of a sphere is $\frac{256\pi}{3}$ cubic centimeters. What is the radius of the sphere, in centimeters? ($V = \frac{4}{3}\pi r^3$) (Grid-in)
3. A rectangle has an area of $48$ square meters. The length of the rectangle is $2$ meters longer than its width. What is the length of the rectangle, in meters?
   • A) $6$
   • B) $8$
   • C) $10$
   • D) $12$
4. The surface area of a cube is $150$ square centimeters. What is the volume of the cube, in cubic centimeters? (Grid-in)
5. A right circular cone has a base area of $16\pi$ and a volume of $48\pi$. What is the height of the cone? ($V = \frac{1}{3}\pi r^2 h$)
   • A) $3$
   • B) $4$
   • C) $9$
   • D) $12$

Level 3: Hard (Advanced SAT)

1. The circumference of a circle is $10\pi$ inches. What is the area of the circle, in square inches?
   • A) $10\pi$
   • B) $20\pi$
   • C) $25\pi$
   • D) $100\pi$
2. Cylinder A and Cylinder B are similar right circular cylinders. The radius of Cylinder B is $2$ times the radius of Cylinder A. How many times greater is the volume of Cylinder B than the volume of Cylinder A? (Grid-in)
3. A right rectangular prism has a square base with a side length of $x$ inches and a height of $5$ inches. If the total surface area of the prism is $112$ square inches, what is the value of $x$? (Grid-in)
4. A rectangular pool is $10$ meters long and $5$ meters wide. The pool is filled with water to a depth of $2$ meters. A solid cube with a side length of $1$ meter is dropped into the pool and sinks completely to the bottom. Assuming no water spills out, by how many meters does the water level in the pool rise?
   • A) $0.02$
   • B) $0.10$
   • C) $0.20$
   • D) $0.50$
5. A triangle has an area of $T$. If the base of the triangle is increased by $20%$ and the height is decreased by $10%$, what is the new area of the triangle in terms of $T$?
   • A) $0.90T$
   • B) $1.08T$
   • C) $1.10T$
   • D) $1.20T$

Level 4: Very Hard (Slightly beyond SAT)

1. A right circular cone and a right circular cylinder have the same radius and the same volume. What is the ratio of the height of the cone to the height of the cylinder? (Grid-in)
2. The diagonal of a square measures $10\sqrt{2}$ units. What is the area of the square, in square units? (Grid-in)
3. A right circular cylinder has a radius of $r$ and a height of $h$. If the radius is increased by $10%$ and the height is decreased by $10%$, by what percentage does the volume of the cylinder increase?
   • A) It decreases by $1%$
   • B) It remains the same
   • C) It increases by $8.9%$
   • D) It increases by $10%$
4. The volume of a solid right pyramid with a square base is $400$ cubic units. If the height of the pyramid is $12$ units, what is the perimeter of the square base?
   • A) $10$
   • B) $20$
   • C) $40$
   • D) $100$
5. A sphere is inscribed perfectly inside a cube such that the sphere touches all six faces of the cube. What is the ratio of the volume of the sphere to the volume of the cube?
   • A) $\frac{\pi}{6}$
   • B) $\frac{\pi}{4}$
   • C) $\frac{\pi}{3}$
   • D) $\frac{2\pi}{3}$

Coordinate geometry, sine and cosine rules and 3D solids

Area and volume · 15 questions

Practice Questions

Level 1: Easy (Foundational)

1. Two cellular towers are located at coordinates $(2, 5)$ and $(10, 11)$ on a city's grid map. A technician needs to install a signal booster exactly halfway between the two towers. What are the coordinates of the signal booster?

   A) $(4, 3)$
   B) $(6, 8)$
   C) $(8, 16)$
   D) $(12, 16)$

2. A skateboard ramp has a horizontal base of $12$ meters and a vertical height of $5$ meters. Determine the sine of the angle of elevation of the ramp from the ground.

   A) $\frac{5}{12}$
   B) $\frac{12}{13}$
   C) $\frac{5}{13}$
   D) $\frac{13}{5}$

3. [Student-Produced Response] A drone takes off from a launch pad at coordinates $(0,0)$ and flies in a straight line to a drop-off point at coordinates $(15, 8)$, where the grid units are in kilometers. What is the total straight-line distance, in kilometers, the drone travelled?

4. A cylindrical water filtration tank has a base radius of $4$ meters and a vertical height of $10$ meters. What is the exact volume of the water tank in cubic meters?

   A) $40\pi$
   B) $80\pi$
   C) $160\pi$
   D) $320\pi$

5. In a Voronoi diagram modelling school districts, the boundary line between School A and School B is the perpendicular bisector of the line segment connecting them. If the line segment connecting School A and School B has a gradient of $\frac{2}{3}$, what is the gradient of the boundary line?

   A) $-\frac{3}{2}$
   B) $-\frac{2}{3}$
   C) $\frac{2}{3}$
   D) $\frac{3}{2}$

Level 2: Medium (Standard)

1. A triangular nature reserve has two boundary fences of lengths $8$ kilometers and $15$ kilometers. The angle between these two fences is $30^\circ$. Determine the exact area of the nature reserve in square kilometers.

   A) $30$
   B) $30\sqrt{3}$
   C) $60$
   D) $60\sqrt{3}$

2. An ant walks inside a hollow rectangular glass box with a length of $3$ cm, a width of $4$ cm, and a height of $12$ cm. What is the length of the straight-line diagonal path through the interior of the box from one bottom corner to the opposite top corner?

   A) $13$ cm
   B) $15$ cm
   C) $17$ cm
   D) $19$ cm

3. [Student-Produced Response] Two tracking ships depart from a naval base at the same time. Ship X travels $5$ miles on a straight path, while Ship Y travels $8$ miles on a different straight path. If the angle between their paths is $60^\circ$, what is the square of the distance between the two ships at their current positions?

4. A student investigates the height of a flagpole. She places a $2$-meter tall vertical stick on the ground, which casts a shadow of $5$ meters. At the same time, the flagpole casts a shadow of $20$ meters. Assuming the ground is perfectly horizontal, determine the height of the flagpole in meters.

   A) $4$
   B) $6$
   C) $8$
   D) $10$

5. [Student-Produced Response] A mechanical windshield wiper rotates through an angle of $60^\circ$ to clean a section of the glass. If the wiper blade sweeps out a circular sector with a radius of $12$ inches, the area of the cleaned sector can be expressed as $k\pi$ square inches. What is the value of $k$?

Level 3: Hard (Advanced)

1. Two competing coffee shops are located at coordinates $A(2, 2)$ and $B(6, 10)$ on a map. A Voronoi boundary line is drawn to divide the delivery zones equally between the two shops. What is the equation of this boundary line?

   A) $y = -0.5x + 8$
   B) $y = 2x - 2$
   C) $y = -0.5x + 10$
   D) $y = 2x + 4$

2. A solid stone pyramid has a square base with a side length of $10$ meters and a vertical height of $12$ meters from the center of the base to the apex. Show that the slant height of a triangular face is $13$ meters, and determine the exact cosine of the angle between a triangular face and the square base.

   A) $\frac{5}{12}$
   B) $\frac{5}{13}$
   C) $\frac{12}{13}$
   D) $\frac{10}{13}$

3. A landscape architect is mapping a triangular piece of land $ABC$. The length of side $a$ is $5\sqrt{3}$ decameters, the length of side $c$ is $5$ decameters, and the angle $C$ is $30^\circ$. Explain why two different shapes for the land are mathematically possible, and determine the larger possible measure for angle $A$.

   A) $60^\circ$
   B) $90^\circ$
   C) $120^\circ$
   D) $150^\circ$

4. [Student-Produced Response] A wooden toy is constructed by attaching a solid hemisphere exactly on top of a solid cylinder of the same radius. The radius of both the cylinder and the hemisphere is $3$ inches, and the vertical height of the cylindrical part is $4$ inches. The total exposed surface area of the toy (which includes the curved part of the hemisphere, the curved side of the cylinder, and the flat circular bottom of the cylinder, but excludes the internal joined faces) is $k\pi$ square inches. What is the value of $k$?

5. A triangular sailing course $PQR$ has side lengths $PQ = x$ km, $PR = (x + 2)$ km, and $QR = 7$ km. If the largest angle in the triangular course is $120^\circ$ and is located at vertex $P$, determine the exact value of $x$.

   A) $2$
   B) $3$
   C) $4$
   D) $5$

Trigonometric identities, vectors and planes

Stretch sheets · 15 questions

Practice Questions

Level 1: Easy (Foundational)

1. An architect calculates the slope of a roof using the secant of an angle of inclination. If the angle is exactly $\frac{\pi}{3}$ radians, determine the exact value of the secant of this angle.
   A) $1$
   B) $\frac{2\sqrt{3}}{3}$
   C) $2$
   D) $\sqrt{3}$
2. A surveying instrument measures an acute angle $\theta$ such that $\tan \theta = \frac{3}{4}$. Using the relevant Pythagorean identity, what is the exact value of $\sec \theta$?
   A) $\frac{3}{5}$
   B) $\frac{4}{5}$
   C) $\frac{5}{4}$
   D) $\frac{5}{3}$
3. In a physics simulation, the work done by a force is the scalar (dot) product of the force vector $\mathbf{F}$ and the displacement vector $\mathbf{d}$. If $\mathbf{F} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and $\mathbf{d} = \begin{pmatrix} 4 \\ 2 \\ -2 \end{pmatrix}$, what is the work done? (SPR)
4. A robotic arm has a joint angle restricted to the principal range of the arcsine function $[-\frac{\pi}{2}, \frac{\pi}{2}]$. If the control software evaluates the expression $\arcsin\left(\sin\left(\frac{5\pi}{6}\right)\right)$, what is the resulting output angle in radians?
   A) $\frac{\pi}{6}$
   B) $\frac{5\pi}{6}$
   C) $-\frac{\pi}{6}$
   D) $\frac{7\pi}{6}$
5. A submarine's displacement from its origin is modelled by the vector $\mathbf{v} = 2\mathbf{i} - 3\mathbf{j} + 6\mathbf{k}$ in kilometres. Investigate the magnitude of this displacement vector to find the straight-line distance from the origin in kilometres.
   A) $5$
   B) $7$
   C) $11$
   D) $49$

Level 2: Medium (Standard)

1. During a light refraction experiment, a laser beam's angle of incidence $\theta$ satisfies $\sin \theta = \frac{1}{3}$. Using a double angle identity, determine the exact value of $\cos(2\theta)$.
   A) $\frac{1}{9}$
   B) $\frac{2}{9}$
   C) $\frac{7}{9}$
   D) $\frac{8}{9}$
2. In a computer graphics rendering engine, the angle between two surface normals must be calculated to apply shading. The normal vectors are $\mathbf{u} = \begin{pmatrix} 1 \\ \sqrt{3} \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} \sqrt{3} \\ 1 \end{pmatrix}$. What is the angle between these two vectors in degrees? (SPR)
3. A mechanical engineer needs to calculate the exact value of $\sin(15^\circ)$ for a gear tolerance analysis. Using the compound angle identity for sine, what is this exact value?
   A) $\frac{\sqrt{6} - \sqrt{2}}{4}$
   B) $\frac{\sqrt{6} + \sqrt{2}}{4}$
   C) $\frac{\sqrt{3} - 1}{2}$
   D) $\frac{\sqrt{2} - 1}{2}$
4. A 3D printing software defines a cutting plane that passes through the point $(2, 1, -1)$ and has a normal vector $\mathbf{n} = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix}$. Which of the following is the Cartesian equation of this plane?
   A) $x - 2y + 3z = 0$
   B) $x - 2y + 3z = -3$
   C) $2x + y - z = 3$
   D) $2x - y + 3z = -3$
5. Show that the area of a solar panel shaped like a parallelogram can be found using the cross product. The adjacent sides are represented by the vectors $\mathbf{a} = 2\mathbf{i}$ and $\mathbf{b} = 3\mathbf{j}$. Determine the area of this solar panel. (SPR)

Level 3: Hard (Advanced)

1. A drone's linear flight path is defined by the vector equation $\mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$. It intersects a vertical netting defined by the plane $x + y + z = 6$. What is the $x$-coordinate of the point of intersection?
   A) $2$
   B) $3$
   C) $4$
   D) $5$
2. Explain the simplification of a sound wave's phase equation, given by the expression $\frac{1 - \cos(2x)}{\sin(2x)}$ for $0 < x < \frac{\pi}{2}$. Which of the following is the fully simplified form?
   A) $\sin x$
   B) $\cos x$
   C) $\tan x$
   D) $\cot x$
3. In a structural analysis, a beam's deflection angle $x$ satisfies $\tan x = \frac{1}{2}$. The critical stress factor is mathematically modelled as $3\tan(2x)$. Determine the exact value of this critical stress factor. (SPR)
4. An electromagnetic simulation calculates the force on a moving charge using the vector product $\mathbf{F} = \mathbf{u} \times \mathbf{v}$. If $\mathbf{u} = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}$, which vector represents $\mathbf{F}$?
   A) $\begin{pmatrix} -3 \\ 3 \\ 3 \end{pmatrix}$
   B) $\begin{pmatrix} 3 \\ -3 \\ 3 \end{pmatrix}$
   C) $\begin{pmatrix} 1 \\ 3 \\ -1 \end{pmatrix}$
   D) $\begin{pmatrix} -1 \\ -3 \\ 3 \end{pmatrix}$
5. An electrical engineer models an alternating current using the complex number $z = 1 + i\sqrt{3}$. Show that $z$ can be written in modulus-argument form and hence determine the exact value of $z^6$.
   A) $-64$
   B) $64$
   C) $64i$
   D) $-64i$

3D vectors, kinematics and the cosine rule

Stretch sheets · 15 questions

Practice Questions

Level 1: Easy (Foundational)

1. A satellite tracks an aircraft in 3D coordinate space. At time $t=1$, the aircraft is at position $A(2, -4, 5)$ and at time $t=3$ it is at $B(8, 6, -1)$. What are the coordinates of the midpoint of the aircraft's path between these two times?

   A) $(3, 1, 2)$
   B) $(5, 1, 2)$
   C) $(5, 2, 3)$
   D) $(6, 10, -6)$

2. A submarine's guidance system uses column vectors to plot courses. The current velocity vector is $\mathbf{u} = \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix}$ and the ocean current vector is $\mathbf{v} = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix}$. What is the resultant vector of $2\mathbf{u} + \mathbf{v}$?

   A) $\begin{pmatrix} 5 \\ 0 \\ 4 \end{pmatrix}$
   B) $\begin{pmatrix} 2 \\ 2 \\ 3 \end{pmatrix}$
   C) $\begin{pmatrix} 7 \\ -8 \\ 0 \end{pmatrix}$
   D) $\begin{pmatrix} 5 \\ 8 \\ 4 \end{pmatrix}$

3. [Student-Produced Response] A drone takes off from a launchpad at the origin $(0,0,0)$ and hovers exactly at the 3D coordinates $(2, 3, 6)$, where all units are in meters. What is the exact straight-line distance, in meters, from the launchpad to the drone?

4. An architect is designing a monument in the shape of a right pyramid with a square base. If the side length of the square base is $6$ meters and the vertical height of the pyramid is $10$ meters, what is the exact volume of the monument in cubic meters?

   A) $60$
   B) $120$
   C) $180$
   D) $360$

5. In a video game, the movement of a character is defined by the vector $\mathbf{a} = \begin{pmatrix} 4 \\ k \end{pmatrix}$, and a laser beam travels along the vector $\mathbf{b} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$. If the character's path is perfectly perpendicular to the laser beam, what is the value of the constant $k$?

   A) $-6$
   B) $-1.5$
   C) $1.5$
   D) $6$

Level 2: Medium (Standard)

1. Determine the exact area of a triangular garden plot that has two sides measuring $8$ meters and $10$ meters, with an included angle of $150^\circ$ between them.

   A) $20$
   B) $20\sqrt{3}$
   C) $40$
   D) $40\sqrt{3}$

2. A surveyor measures a triangular parcel of land. Two sides measure $3$ kilometers and $8$ kilometers, and the angle between these two sides is $60^\circ$. What is the exact length of the third side of the parcel in kilometers?

   A) $7$
   B) $8$
   C) $\sqrt{73}$
   D) $\sqrt{97}$

3. [Student-Produced Response] A drone moves at a constant velocity, and its position vector $\mathbf{r}$ at time $t$ seconds is given by the kinematic equation $\mathbf{r} = \begin{pmatrix} 2 \\ 5 \end{pmatrix} + t \begin{pmatrix} 3 \\ -4 \end{pmatrix}$ in meters. What is the exact speed of the drone in meters per second?

4. In a Voronoi diagram mapping out competing delivery zones, the boundary between Store A at $(1, 1)$ and Store B at $(5, 5)$ is the perpendicular bisector of the line segment connecting them. What is the equation of this boundary line?

   A) $y = x$
   B) $y = -x + 6$
   C) $y = -x + 8$
   D) $y = 0.5x + 1.5$

5. [Student-Produced Response] A paperweight is shaped like a solid hemisphere with a radius of $4$ centimeters. The total surface area of the paperweight, including its flat circular base, is expressed as $k\pi$ square centimeters. What is the value of $k$?

Level 3: Hard (Advanced)

1. Two aircraft are flying along straight-line paths described by vector equations. Aircraft 1 has position $\mathbf{r}_1 = \begin{pmatrix} 0 \\ 10 \end{pmatrix} + t \begin{pmatrix} 4 \\ -2 \end{pmatrix}$ and Aircraft 2 has position $\mathbf{r}_2 = \begin{pmatrix} 12 \\ 4 \end{pmatrix} + t \begin{pmatrix} 1 \\ c \end{pmatrix}$, where $t$ is time in hours. If the two aircraft collide (reach the same position at the exact same time), what must be the value of the constant $c$?

   A) $-1$
   B) $-0.5$
   C) $0.5$
   D) $1$

2. Investigate the geometric relationship between the 3D vectors $\mathbf{u} = \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix}$. By computing their scalar product, determine the exact cosine of the angle $\theta$ between them.

   A) $\frac{8}{9}$
   B) $\frac{7}{9}$
   C) $\frac{2}{3}$
   D) $\frac{\sqrt{5}}{3}$

3. In triangle $ABC$, side $a = 10$, side $b = 10\sqrt{3}$, and angle $A = 30^\circ$. Explain why the ambiguous case occurs, and determine the two possible mathematical values for angle $B$.

   A) $30^\circ$ and $150^\circ$
   B) $60^\circ$ and $120^\circ$
   C) $45^\circ$ and $135^\circ$
   D) Only $60^\circ$ is possible.

4. A glass paperweight is a solid square-based pyramid. The side length of the square base is $10$ cm, and the vertical height of the pyramid (from the center of the base to the apex) is $12$ cm. Show that the slant height of a triangular face is $13$ cm, and determine the exact length of one of the slanted edges connecting a base corner to the apex.

   A) $\sqrt{169}$
   B) $\sqrt{194}$
   C) $\sqrt{269}$
   D) $15$

5. [Student-Produced Response] A high-speed particle moves through a 3D magnetic field along a linear path defined by the vector equation $\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix}$. At a certain instant, the particle passes through a sensor located at a point $P$ on this line. If the $y$-coordinate of point $P$ is $-5$, what is the exact $z$-coordinate of point $P$?