∆PQR has vertices at P(2, 4), Q(3, 8) and R(5, 4). A dilation and series of translations map ∆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?
To find the scale factor of the dilation in the similarity transformation, we can compare the corresponding side lengths of the two triangles.
First, let's find the side lengths of triangle PQR:
- The length of PQ can be found using the distance formula: √[(x2 - x1)^2 + (y2 - y1)^2].
So, PQ = √[(3 - 2)^2 + (8 - 4)^2] = √[1^2 + 4^2] = √17.
- The length of QR can be found similarly: QR = √[(5 - 3)^2 + (4 - 8)^2] = √[2^2 + 4^2] = √20.
- The length of RP can be found similarly: RP = √[(2 - 5)^2 + (4 - 4)^2] = √[(-3)^2 + 0^2] = √9 = 3.
Now, let's find the corresponding side lengths of triangle ABC:
- The length of AB can be found similarly: AB = √[(5.5 - 2)^2 + (18 - 4)^2] = √[3.5^2 + 14^2] = √12.25 + 196 = √208.25.
- The length of BC can be found similarly: BC = √[(12.5 - 5.5)^2 + (4 - 18)^2] = √[7^2 + (-14)^2] = √49 + 196 = √245.
- The length of CA can be found similarly: CA = √[(2 - 12.5)^2 + (4 - 4)^2] = √[(-10.5)^2 + 0^2] = √110.25.
Now, we can find the scale factor by comparing the corresponding side lengths of the triangles:
- Scale factor = (Length of AB)/(Length of PQ) = (√208.25)/(√17) = √12.25 ≈ 3.5.
Therefore, the scale factor of the dilation in the similarity transformation is approximately 3.5.