Question

To derive the standard formula of a circle, a radius r is constructed as the hypotenuse of a right triangle as shown.

On a coordinate plane, a circle has center D at point (h, k). Right triangle D F E has point E on the circle. Point E is at (x, y). The length of D E is r. Angle E F D is a right angle.

What are the lengths of the sides of the right triangle that is formed? Check all that apply.

h
k
h – k
x – h
x – y
y – h
y – k

Answer 1

The legs are the horizontal and vertical distances from E(x,y) to the center D(h,k), i.e. |x − h| and |y − k| (listed as x − h and y − k). The hypotenuse is r, giving (x − h)^2 + (y − k)^2 = r^2.

The other listed differences (h, k, h − k, x − y, y − h) are not the triangle's side lengths.