Prove that the two circles shown below are similar. circle x is shown with a center at negative 2, 8 and a radius of 6. circle y is shown with a center of 4, 2 and a radius of 3.

They are similar because the radius of circle x is a multiple of circle y. proof: translate smaller circle y to the location of circle x  so that they are concentric (same center location (2, the translation would be moving the center of y left 2 units and up 6 units. now that the centers are concentric a dilation is need to increase the size of circle y to coincide with circle x. dilation by the scale factor, k k = 6/3 k = 2 as you can see the scale factor, k is a ratio of the radius thus proving the circles are similar

Given

ΔABC with ∠A = 90°ΔADC with ∠D = 90°The two triangles share same angle ∠C

With two same angles the triangles are similar according to AA similarity postulate:

ΔABC ~ ΔDAC

As we know if triangles are similar then their corresponding sides have same ratio:

AC/CD = BC/AC ⇒ AC*AC = BC*CD ⇒ AC² = BC*CD