Question

100 Points! Geometry question. Photo attached. Please show as much work as possible. Thank you!

Answer 1

`(x - 3)^2 + (y + 2)^2 = 16`

Explanation:

The equation of a circle with a center at (h, k) and radius r is given by:

`(x - h)^2 + (y - k)^2 = r^2`

We are given that the points G(5,-2) and H(1, −2) lie on the circle. We can use the distance formula to find the distance between these two points.

`d = √((x_1 - x_2)^2 + (y_1 - y_2)^2)`

`d = √((5 - 1)^2 + ((-2) - (-2))^2) = √(16) = 4`

Therefore, the radius of the circle is 4.

We can now find the center of the circle by taking the average of the x-coordinates and y-coordinates of the points G and H.

`(h, k) = \left((x_1 + x_2)/(2), (y_1 + y_2)/(2)\right) = \left((5 + 1)/(2), (-2 - 2)/(2)\right) = (3, -2)`

Therefore, the equation of the circle is:

`(x - 3)^2 + (y + 2)^2 = 4^2`

Simplifying the equation, we get:

`\bold{(x - 3)^2 + (y + 2)^2 = 16}` is a required equation.