Question

1. below is a dilation with center p of triangle abc resulting into triangle abc. a. what is the scale factor of the dilation? b. which segment is parallel to segment ac? measure of angle c is 60 degrees, what are the following

Explanation:

Step1: Identify corresponding sides

To find the scale factor, we use corresponding sides of the original triangle (ABC) and the dilated triangle (A'B'C'). Let's take sides \( AB = 14 \) and \( A'B' = 7 \), or \( AC = 10 \) and \( A'C' = 5 \), or \( BC = 21 \) and \( B'C' = 10.5 \).

Step2: Calculate the scale factor

The scale factor \( k \) is the ratio of the length of a side in the dilated triangle to the length of the corresponding side in the original triangle. Using \( AB = 14 \) and \( A'B' = 7 \):
\( k=\frac{A'B'}{AB}=\frac{7}{14}=\frac{1}{2} \).
We can verify with another pair, e.g., \( AC = 10 \) and \( A'C' = 5 \): \( \frac{5}{10}=\frac{1}{2} \), or \( BC = 21 \) and \( B'C' = 10.5 \): \( \frac{10.5}{21}=\frac{1}{2} \).

Step3: Determine parallel segments (for part B)

In a dilation, corresponding sides are parallel. So, segment \( A'B' \) is parallel to \( AB \), and segment \( A'C' \) is parallel to \( AC \). But the question asks which segment is parallel to \( A'C' \). From the diagram, \( AC \) is the corresponding side to \( A'C' \), so \( AC \) is parallel to \( A'C' \).

Answer:

(for part A):
The scale factor of the dilation is \(\frac{1}{2}\) (or 0.5).