Question

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 PQR and P'Q'R' on your own coordinate grid.

Part A: What is the scale factor of the dilation that transforms triangle PQR to triangle P'Q'R'? Explain your answer.

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

Answer 1

The scale factor of the dilation that transforms triangle PQR to triangle P'Q'R' is approximately 0.5104. When triangle P'Q'R' is reflected about the y-axis, the coordinates of the reflected triangle P''Q''R'' are P''( -1, -2), Q''(0, 3), R''(1, 0).

Step-by-step explanation:

Part A: To find the scale factor of the dilation that transforms triangle PQR to triangle P'Q'R', we can compare the lengths of corresponding sides. We'll focus on one side, PQ, and its corresponding side P'Q'.

The length of PQ is found using the distance formula:

d(P, Q) = √((x2 - x1)^2 + (y2 - y1)^2)

d(P, Q) = √((0 - 3)^2 + (9 - ( -6))^2)

d(P, Q) = √(9 + 225)

d(P, Q) = √234

The length of P'Q' can be found using the same formula:

d(P', Q') = √((0 - 1)^2 + (3 - ( -2))^2)

d(P', Q') = √(1 + 25)

d(P', Q') = √26

To find the scale factor, we divide the length of P'Q' by the length of PQ:

Scale factor = d(P', Q') / d(P, Q)

Scale factor = √26 / √234

Scale factor ≈ 0.5104

Part B: To reflect the coordinates of P'Q'R' about the y-axis, we negate the x-coordinate while keeping the y-coordinate the same. So, the coordinates of the reflected triangle P''Q''R'' would be:

P''( -1, -2), Q''(0, 3), R''(1, 0)