Worked Examples: SAT Strategy
Example 1: Easy (Proportions for Arcs and Sectors)
Question: A circle has a circumference of $36\pi$. An arc on the circle has a length of $4\pi$. What is the measure, in degrees, of the central angle that creates this arc?
SAT Strategy: Do not try to memorize complex formulas for arcs and sectors. Just remember that everything in a circle is a perfect proportion: $\frac{\text{Angle}}{360^\circ} = \frac{\text{Arc Length}}{\text{Circumference}} = \frac{\text{Sector Area}}{\text{Total Area}}$.
- Set up the proportion: $\frac{\text{Angle}}{360} = \frac{4\pi}{36\pi}$.
- Simplify the right side: The $\pi$ cancels out, and $\frac{4}{36} = \frac{1}{9}$.
- Solve for the Angle: $\frac{\text{Angle}}{360} = \frac{1}{9} \implies \text{Angle} = \frac{360}{9} = 40^\circ$.
- Answer: 40.
Example 2: Medium (The Circle Equation)
Question: A circle in the $xy$-plane has the equation $(x - 4)^2 + (y + 19)^2 = 121$. What is the least possible $x$-coordinate of any point on the circle?
SAT Strategy: Extract the center $(h, k)$ and the radius $r$ directly from the standard form $(x-h)^2 + (y-k)^2 = r^2$.
- The center $(h, k)$ has the opposite signs of the numbers in the parentheses: Center $= (4, -19)$.
- The radius is the square root of the constant on the right: $r = \sqrt{121} = 11$.
- The "least possible $x$-coordinate" represents the leftmost edge of the circle. Start at the center's $x$-coordinate ($4$) and move left by exactly one radius ($11$).
- $x = 4 - 11 = -7$.
- Answer: -7.
Example 3: Hard (Completing the Square)
Question: The equation of a circle in the $xy$-plane is $x^2 + y^2 - 10x + 6y = 2$. What is the radius of the circle?
SAT Strategy: When a circle's equation is expanded, you must "complete the square" for both $x$ and $y$ to group them back into standard form and reveal the true radius squared ($r^2$).
- Group the $x$ terms and $y$ terms: $(x^2 - 10x) + (y^2 + 6y) = 2$.
- Find the magic numbers to add. Take half of the middle coefficient and square it:
- For $x$: half of $-10$ is $-5$, squared is $25$.
- For $y$: half of $6$ is $3$, squared is $9$.
- Add these numbers to both sides of the equation: $(x^2 - 10x + 25) + (y^2 + 6y + 9) = 2 + 25 + 9$.
- Factor the left side and simplify the right side: $(x - 5)^2 + (y + 3)^2 = 36$.
- The right side represents $r^2$, so $r^2 = 36 \implies r = 6$.
- Answer: 6.
Practice Questions
Level 1: Easy (Foundational SAT)
- 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) $x^2 - y^2 = 64$
- 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) $(-5, -3)$
- 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) $36\pi$
- 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)
- An angle measures $\frac{\pi}{4}$ radians. What is the measure of this angle in degrees? (Grid-in)
Level 2: Medium (Standard SAT)
- 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) $(x + 3)^2 + (y + 4)^2 = 25$
- The equation of a circle is $x^2 + y^2 + 8x - 2y = 8$. What is the radius of the circle? (Grid-in)
- 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)
- 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) $(-2, 12)$
- 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)
- 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) $36$
- 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)
- 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) $(x + 8)^2 + (y - 3)^2 = 64$
- 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)
- 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)
- 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) $96\pi$
- 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) $\frac{4}{3}$
- 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)
- 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)
- 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)
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