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