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.
Part C: Are the two triangles PQR and P''Q''R'' congruent? Explain your answer.
To answer these questions, we will first plot the given points on a coordinate grid.
Plotting the points for Triangle PQR:
P(3, -6)
Q(0, 9)
R(-3, 0)
Plotting the points for Triangle P'Q'R':
P'(1, -2)
Q'(0, 3)
R'(-1, 0)
Now, let's move on to each part of the question.
Part A: What is the scale factor of the dilation that transforms triangle PQR to triangle P'Q'R'? Explain your answer.
To find the scale factor of the dilation, we need to compare the corresponding side lengths of the two triangles.
From the given coordinates, we can calculate the side lengths for Triangle PQR:
Side PQ: √((0-3)² + (9-(-6))²) = √(9 + 225) = √234
Side QR: √((-3-0)² + (0-9)²) = √(9 + 81) = √90
Side RP: √((3-(-3))² + (-6-0)²) = √(36 + 36) = √72
Similarly, we can find the side lengths for Triangle P'Q'R':
Side P'Q': √((0-1)² + (3-(-2))²) = √(1 + 25) = √26
Side Q'R': √((-1-0)² + (0-3)²) = √(1 + 9) = √10
Side R'P': √((1-(-1))² + (-2-0)²) = √(4 + 4) = √8
Now, we can find the scale factor by comparing the corresponding side lengths:
Scale factor = (P'Q' / PQ) = (√26 / √234) ≈ 0.197
Therefore, the scale factor of the dilation that transforms triangle PQR to triangle P'Q'R' is approximately 0.197.
Part B: Write the coordinates of triangle P"Q"R" obtained after P'Q'R' is reflected about the y-axis.
Reflecting a point about the y-axis negates the x-coordinate. So, to obtain the coordinates of triangle P"Q"R", we simply need to negate the x-coordinate of each point in triangle P'Q'R':
P"(-1, -2)
Q"(0, 3)
R"(1, 0)
Therefore, triangle P"Q"R" obtained after P'Q'R' is reflected about the y-axis has vertices P"(-1, -2), Q"(0, 3), and R"(1, 0).
Part C: Are the two triangles PQR and P''Q''R'' congruent? Explain your answer.
To determine if the two triangles PQR and P''Q''R'' are congruent, we need to check if their corresponding angles and side lengths are equal.
Angles:
The angles of the two triangles will remain the same since reflecting a triangle does not alter its angles.
Side lengths:
Comparing the side lengths of the two triangles:
PQ ≈ √234
Q'R" ≈ √26
QR ≈ √90
PR ≈ √72
P"R" ≈ √8
RQ ≈ √90
R"P" ≈ √26
Considering both the angles and the side lengths, it is clear that the triangles PQR and P''Q''R'' are not congruent, as their side lengths are different.
Therefore, the triangles PQR and P''Q''R'' are not congruent.
A: 1/3
reflection across y takes (x,y)→(-x,y)
dilation destroyed the congruency