Triangle PQR is transformed to triangle P'Q'R. Triangle PQR has vertices P (3, -6), Q (0, -9), and R (-3,0). Triangle P'Q'R' has vertices P' (1, -2), Q' (0. 3), and R' (1,0). Plot triangles and P'Q'R' on your own coordinate grid. A. What is the scale factor of the dilation that transform triangle PQR to triangle P'Q'R? Explain your answer.

B. Write the coordinates of triangle P"Q"R obtained after P'Q'R is reflected about the y-axis.

C. Are the two triangles PQR and P"Q"R congruent? Explain your answer.

Please see the attached plot of triangle P'Q'R'

A. The scale factor is 1/3

B. The coordinates of the triangle P''Q''R'' obtained after P'Q'R' is reflected about the y-axis are p''(-1, -2), Q''(0, -3), and R''(1, 0)

C. The two triangles PQR and P''Q''R'' are not congruent

Step-by-step explanation:

A. The coordinates of triangle PQR are P(3, -6), Q(0, -9) and R(-3, 0)

The vertices of triangle P'Q'R' are corrected to be P'(1, -2), Q'(0, -3), and R'(-1, 0)  

The length, l, of the segment can be calculated as follows;

The lengths of the sides of triangle PQR are;

Using an online application, we get;

Length of PQ = √18 = 3·√2

Length of PR = √72 = 6·√2

Length of RQ = √90 = 3·√10

The lengths of the sides of triangle P'Q'R' are;

Length of P'Q' = √2

Length of P'R' = √8 = 2·√2

Length of R'Q' = √10  

We have; PQ/(P'Q') = 3·√2/(√2) = 3

PR/(P'R') = 6·√2/(2·√2) = 3

RQ/(R'Q') = 3·√10/√10 = 3

The lengths of the sides of triangle P'Q'R' are 1/3 times the length of triangle PQR, therefore, the scale factor = 1/3

B. For reflection  about the y-axis, we have the coordinates of the preimage as (x, y) and the coordinates of the image as (-x, y);

Therefore, we have;

P'(1, -2) after reflection  about the y-axis becomes p''(-1, -2)

Q'(0, -3) after reflection  about the y-axis becomes Q''(0, -3)

R'(-1, 0)  after reflection  about the y-axis becomes R''(1, 0)

The coordinates of the triangle P''Q''R'' are p''(-1, -2), Q''(0, -3), and R''(1, 0)

C. Triangle P''Q''R'' is a reflection of triangle PQR, which is a rigid transformation. Therefore, the lengths of the sides of PQR and P''Q''R'' are the same. However, triangle P'Q'R' is a dilation of triangle PQR by a scale factor of 1/3 therefore, triangle P''Q''R'' is not congruent to triangle PQR.