Question

Right triangle ABC is located at A (−1, 4), B (−1, 1), and C (−5, 1) on a coordinate plane. What is the equation of a circle A with radius segment AC?

(x + 1)2 + (y − 4)2 = 9
(x + 5)2 + (y − 1)2 = 25
(x + 5)2 + (y − 1)2 = 16
(x + 1)2 + (y − 4)2 = 25

Answer 1

To find the equation of circle A with radius equal to segment AC, we first need to determine the length of segment AC.

Coordinates of points:
- A = (−1, 4)
- C = (−5, 1)

We can use the distance formula to find the length of segment AC:

\[
AC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
\[
AC = \sqrt{((-5) - (-1))^2 + (1 - 4)^2}
\]
\[
= \sqrt{(-5 + 1)^2 + (1 - 4)^2}
\]
\[
= \sqrt{(-4)^2 + (-3)^2}
\]
\[
= \sqrt{16 + 9}
\]
\[
= \sqrt{25}
\]
\[
= 5
\]

The length of segment AC is 5.

Now, the equation of a circle centered at point A (−1, 4) with radius 5 is given by the formula:

\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center and \(r\) is the radius.

In our case:
- Center (h, k) = (−1, 4)
- Radius \(r = 5\)

So we plug in these values into the equation:

\[
(x + 1)^2 + (y - 4)^2 = 5^2
\]
\[
(x + 1)^2 + (y - 4)^2 = 25
\]

Thus, the equation of circle A is:

\[
(x + 1)^2 + (y - 4)^2 = 25
\]

The correct answer is:

**(x + 1)² + (y - 4)² = 25**.