Geometry and Trigonometry: Lines, Angles, and Triangles

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) $5$