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

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

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.