Question

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?

Answer 1

Answer with explanation:

⇒Vertices of Δ PQR = P(2, 4), Q(3, 8) and R(5, 4)

Area of Δ PQR

           $=\frac{1}{2} \times \left[\begin{array}{ccc}2&4&1\\3&8&1\\5&4&1\end{array}\right] \\\\=\frac{1}{2} \times[ 2\times(8-4)-4 \times (3-5)+1 \times (12-40)]\\\\=\frac{1}{2} \times [8+8-28]\\\\=\frac{1}{2} \times|16-28|\\\\=\frac{1}{2} \times 12\\\\=6 \text{square units}$

⇒Vertices of Δ ABC  A(2, 4), B(5.5, 18), and C(12.5, 4).

Area of Δ  ABC

           $=\frac{1}{2} \times \left[\begin{array}{ccc}2&4&1\\5.5&18&1\\12.5&4&1\end{array}\right] \\\\=\frac{1}{2} \times[ 2\times(18-4)-4\times (5.5-12.5)+1 \times (22-225)]\\\\=\frac{1}{2} \times [28+28-203]\\\\=\frac{1}{2} \times|56-203|\\\\=\frac{1}{2} \times 147\\\\=73.50\text{square units}$

⇒Scale Factor or Dilation in this transformation

 = is Area of Image Divided by Area of Preimage

         $=\frac{\text{Area of} \Delta ABC}{\text{Area of} \Delta PQR}\\\\=\frac{73.50}{6}\\\\=12.25$

       

Answer 2

D is the correct answer.