In the -plane, a circle passes through the points , , and . Which choice is the area of the circle?

Solution available with step-by-step explanation

Super-Hard SAT Math Question 79 | Free SAT Prep | Aniko

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Question 79·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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In the $xy$ -plane, a circle passes through the points $(1,1)$ , $(5,1)$ , and $(3,5)$ .

Which choice is the area of the circle?

For a coordinate-circle question determined by several points, first use symmetry or perpendicular bisectors to narrow down the center. Then set two distances equal to solve for the center, and use that center to compute the radius. This keeps the work organized and avoids guessing the circle equation from the start.

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Start with the horizontal chord

Since $(1,1)$ and $(5,1)$ have the same $y$ -coordinate, the midpoint of that segment is $(3,1)$ , so the center must lie on the vertical line $x=3$ .

Use equal distances

If the center is $(3,k)$ , its distance to $(1,1)$ must equal its distance to $(3,5)$ . Set the squared distances equal.

Desmos Guide

Represent the unknown y-coordinate of the center

Since the center lies on $x=3$ , let its coordinates be (3,k). In Desmos, graph y = 4 + (x - 1)^2 and y = (x - 5)^2, using x as the value of k. Their intersection gives the y-coordinate of the center.

Compute r^2 and the area

Use the intersection value k = 5/2, then enter r2 = (3-1)^2 + (5/2 - 1)^2 and A = pi*r2. The value of A matches the correct answer choice.

Step-by-step Explanation

Locate the center on a perpendicular bisector

The points $(1,1)$ and $(5,1)$ form a horizontal chord. Its midpoint is $(3,1)$ , so the perpendicular bisector is the vertical line $x=3$ . Let the center be $(3,k)$ .

Solve for the center

Because all radii are equal,

(3-1)^2+(k-1)^2=(3-3)^2+(k-5)^2.

4+(k-1)^2=(k-5)^2.

Expanding gives

4+k^2-2k+1=k^2-10k+25,

so $8k=20$ and $k=\frac{5}{2}$ .

Find the area

Now compute $r^2$ using the center $\left(3,\frac{5}{2}\right)$ and the point $(1,1)$ :

r^2=(3-1)^2+\left(\frac{5}{2}-1\right)^2 =4+\left(\frac{3}{2}\right)^2 =\frac{25}{4}.

Therefore, the area is

\pi r^2=\frac{25\pi}{4}.

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation