Question

A triangle with vertices at A(0, 0), B(80, 4), and C(6, 0) is dilated to yield a triangle with vertices at A(0, 0), B(0, 10), and C(15, 0). The origin is the center of dilation. What is the scale factor of the dilation?

A. 1.5
B. 2
C. 2.5
D. 3

Answer 1

The scale factor of the dilation is approximately 2.2047.

Step-by-step explanation:

To find the scale factor of the dilation, we need to compare the corresponding side lengths of the original triangle and the dilated triangle. Let's start with side AB. The length of AB in the original triangle is found using the distance formula: AB = √((80-0)^2 + (4-0)^2) = √(80^2 + 4^2) = √(6400 + 16) = √(6416).

The length of AB in the dilated triangle is found using the distance formula: AB' = √((0-0)^2 + (10-0)^2) = √(0^2 + 10^2) = √(100) = 10. Therefore, the scale factor for side AB is AB'/AB = 10/√(6416).

Similarly, we can find the scale factor for side BC. BC = √((6-80)^2 + (0-4)^2) = √((-74)^2 + (-4)^2) = √(5476 + 16) = √(5492). BC' = √((15-0)^2 + (0-0)^2) = √(15^2) = 15. So the scale factor for side BC is BC'/BC = 15/√(5492).

Since the dilated triangle is an enlargement of the original triangle, the scale factor should be greater than 1. Evaluating the scale factors, we find that AB'/AB = 10/√(6416) is approximately 2.1349 and BC'/BC = 15/√(5492) is approximately 2.2047. Therefore, the scale factor of the dilation is approximately 2.2047, which corresponds to option C.