Question

⊙O and ⊙P are given with centers (−2, 7) and (12, −1) and radii of lengths 5 and 12, respectively. Using similarity transformations on ⊙O, prove that ⊙O and ⊙P are similar.

Answer 1

Answer:

Whereby circle $\bigodot$P can be obtained from circle $\bigodot$O by applying the transformations of a translation of T₍₁₄, ₋₈₎ followed by a dilation by a scale factor of 2.4, $\bigodot$O is similar to $\bigodot$P

Step-by-step explanation:

The given center of the circle $\bigodot$O = (-2, 7)

The radius of $\bigodot$O, r₁ = 5

The given center of the circle $\bigodot$P = (12, -1)

The radius of $\bigodot$P, r₂ = 12

The similarity transformation to prove that $\bigodot$O and $\bigodot$P are similar are;

a) Move circle $\bigodot$O 14 units to the right and 8 units down to the point (12, -1)

b) Apply a scale of S.F. = r₂/r₁ = 12/5 = 2.4

Therefore, the radius of circle $\bigodot$O is increased by 2.4

We then obtain $\bigodot$O' with center at (12, -1) and radius r₃ = 2.4×5 = 12 which has the same center and radius as circle $\bigodot$P

∴ Circle $\bigodot$P can be obtained from circle $\bigodot$O by applying similarity transformation of translation of T₍₁₄, ₋₈₎ followed by a dilation by a scale factor of 2.4, $\bigodot$O is similar to $\bigodot$P.