PQR has vertices at P (2, 4), Q (3, 8) and R (5, 4). A similarity transformation maps PQR to ABC, whose vertices are A (2, 4), B (5.5, 18), and C (12.5, 4). What is the scale factor of the dilation in the similarity transformation?

Use Pythagorean Theorem to calculate length of sides

PQ = √ (1^2 + 4^2) = √ (17)

QR = √ (2^2 + 4^2) = √ (20)

RP = √ (3^2 + 0^2) = √ (9)

AB = √ (3.5^2 + 14^2) = √ (208.25)

BC = √ (7^2 + 14^2) = √ (245)

CA = √ (10.5^2 + 0^2) = √ (110.25)

A similarity transformation will maintain the relationship of sides: the smallest side of one triangle should correspond to the shortest side of the other triangle (and so on).

Ratio of lengths (transformed/original)

shortest with shortest

CA/RP = √ (110.25) / √ (9) = 10.5/3 = 3.5

middle

AB/PQ = √ (208.25) / √ (17) = √ (208.25/17) = √ (12.25) = 3.5

longest

BC/QR = √ (245) / √ (20) = √ (12.25) = 3.5