Question

Problem 1) Find the scale factor of dilation for the triangles below.
the coordinates of P (1,3), R (1,1), Q (4,1).
the coordinates of P' (3,8), R' (3,4), Q' (9,4)
the answers are:
k = 1
k = 1

k = 2
k = 2

k = 5
k = 5

k = 8

Answer 1

To find the scale factor of dilation for the triangles, we need to calculate the lengths of corresponding sides of the original triangle and the dilated triangle and determine the ratio between these lengths.

First, let's find the lengths of the sides for triangle PQR:
1. The distance between points P (1,3) and R (1,1):
\[
PR = \sqrt{(1-1)^2 + (3-1)^2} = \sqrt{0 + 2^2} = \sqrt{4} = 2
\]

2. The distance between points R (1,1) and Q (4,1):
\[
RQ = \sqrt{(1-4)^2 + (1-1)^2} = \sqrt{(-3)^2 + 0} = \sqrt{9} = 3
\]

3. The distance between points Q (4,1) and P (1,3):
\[
PQ = \sqrt{(1-4)^2 + (3-1)^2} = \sqrt{(-3)^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}
\]

Now, for triangle P'R'Q':
1. The distance between points P' (3,8) and R' (3,4):
\[
P'R' = \sqrt{(3-3)^2 + (8-4)^2} = \sqrt{0 + 4^2} = \sqrt{16} = 4
\]

2. The distance between points R' (3,4) and Q' (9,4):
\[
R'Q' = \sqrt{(3-9)^2 + (4-4)^2} = \sqrt{(-6)^2 + 0} = \sqrt{36} = 6
\]

3. The distance between points Q' (9,4) and P' (3,8):
\[
P'Q' = \sqrt{(3-9)^2 + (8-4)^2} = \sqrt{(-6)^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13}
\]

Now we can calculate the scale factor \( k \) for each corresponding side of the triangles:

1. For side PR and P'R':
\[
k = \frac{P'R'}{PR} = \frac{4}{2} = 2
\]

2. For side RQ and R'Q':
\[
k = \frac{R'Q'}{RQ} = \frac{6}{3} = 2
\]

3. For side PQ and P'Q':
\[
k = \frac{P'Q'}{PQ} = \frac{2\sqrt{13}}{\sqrt{13}} = 2
\]

In all cases, we find the scale factor \( k \) is the same.

The scale factor of dilation \( k \) for the triangles is:
\[
\boxed{2}
\]