A circular stage is shown on an -coordinate grid. The center of the stage is at , and a point on the edge of the stage is at . Which choice is the area of the stage, in square units, in terms of ?

Solution available with step-by-step explanation

SAT Circles Question 39 | Free SAT Prep | Aniko

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Question 39·Medium·Circles

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A circular stage is shown on an $xy$ -coordinate grid. The center of the stage is at $C(2,-1)$ , and a point on the edge of the stage is at $P(8,5)$ .

Which choice is the area of the stage, in square units, in terms of $\pi$ ?

For circle questions on the coordinate plane, use the distance formula to get the radius (or directly compute $r^2$ ). Then plug into the appropriate circle formula (here, $A=\pi r^2$ ) and match the result to the choices.

Hints

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Identify the radius

A radius is the segment from the center of a circle to any point on the circle.

Use coordinates to find the radius

Use the distance formula between $C(2,-1)$ and $P(8,5)$ . You can compute $r^2$ without taking a square root.

Convert radius to area

Once you have $r^2$ , use $A=\pi r^2$ .

Desmos Guide

Compute $r^2$ directly

In Desmos, enter (8-2)^2 + (5-(-1))^2 to compute $r^2$ .

Use $A=\pi r^2$

Use the value of $r^2$ you found to write the area as $A=\pi r^2$ (the coefficient of $\pi$ is $r^2$ ).

Match to a choice

Since the coefficient of $\pi$ is $72$ , the matching choice is $72\pi$ .

Step-by-step Explanation

Find the radius using the distance formula

Because $C(2,-1)$ is the center and $P(8,5)$ is on the circle, $CP$ is a radius. Use the distance formula:

r^2=(8-2)^2+(5-(-1))^2.

Compute $r^2$

Compute each difference and square:

r^2=6^2+6^2=36+36=72.

Use the area formula

The area of a circle is $A=\pi r^2$ . With $r^2=72$ ,

A=\pi(72)=72\pi.

So, the area of the stage is $72\pi$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation