Please help
Triangle PQR is transformed to triangle P'Q'R'. Triangle PQR has vertices P(8, 0), Q(6, 2), and R(−2, −4). Triangle P'Q'R' has vertices P'(4, 0), Q'(3, 1), and R'(−1, −2).
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. (4 points)
Part B: Write the coordinates of triangle P"Q"R" obtained after P'Q'R' is reflected about the y-axis. (4 points)
Part C: Are the two triangles PQR and P''Q''R'' congruent? Explain your answer. (2 points)
y'all please help I'm struggling here
(A) to find the scale factor, compare the side lengths. That is, P'Q'/PQ
(B) (x,y) → (-x,y)
(C) unless the scale factor is 1, the triangles are similar, not congruent. The reflection does not change that. P'Q'R' ≅ P"Q"R"
To plot triangles PQR and P'Q'R' on a coordinate grid, we can use a graphing software or draw it manually. Here is a step-by-step solution to each part of the question:
Part A: To find the scale factor of the dilation that transforms triangle PQR to triangle P'Q'R', you need to find the ratio of the corresponding side lengths.
First, calculate the length of each side of triangle PQR:
Side PQ:
√[(6 - 8)^2 + (2 - 0)^2] = √4 + 4 = √8 = 2√2
Side QR:
√[(-2 - 6)^2 + (-4 - 2)^2] = √64 + 36 = √100 = 10
Side RP:
√[(8 - (-2))^2 + (0 - (-4))^2] = √100 + 16 = √116 ≈ 10.77
Next, calculate the length of each corresponding side of triangle P'Q'R':
Side P'Q':
√[(3 - 4)^2 + (1 - 0)^2] = √1 + 1 = √2 ≈ 1.41
Side Q'R':
√[(-1 - 3)^2 + (-2 - 1)^2] = √16 + 9 = √25 = 5
Side R'P':
√[(4 - (-1))^2 + (0 - (-2))^2] = √25 + 9 = √34 ≈ 5.83
Now, find the ratio of the corresponding side lengths:
Scale factor = (P'Q' / PQ) = 1.41 / (2√2) ≈ 0.70
Therefore, the scale factor of the dilation is approximately 0.70. This means that triangle P'Q'R' is scaled down by a factor of 0.70 compared to triangle PQR.
Part B: To reflect triangle P'Q'R' about the y-axis, negate the x-coordinates while keeping the y-coordinates unchanged. This is because reflecting about the y-axis involves switching the sign of the x-coordinates.
Triangle P"Q"R" has the following coordinates:
P"(−4, 0)
Q"(−3, 1)
R"(1, −2)
Part C: The two triangles PQR and P''Q''R'' are not congruent. Congruent triangles have exactly the same size and shape. Since triangle PQR has been transformed by a dilation and reflected about the y-axis to obtain triangle P"Q"R", their sizes and shapes are different.
Part A: To find the scale factor of the dilation that transforms triangle PQR to triangle P'Q'R', we need to compare the corresponding side lengths of the two triangles.
The distance between two points can be found using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
For triangle PQR, we can calculate the lengths of its sides:
- Side PQ: sqrt((6 - 8)^2 + (2 - 0)^2) = sqrt(4 + 4) = sqrt(8)
- Side QR: sqrt((-2 - 6)^2 + (-4 - 2)^2) = sqrt(64 + 36) = sqrt(100) = 10
- Side RP: sqrt((-2 - 8)^2 + (-4 - 0)^2) = sqrt(100 + 16) = sqrt(116)
For triangle P'Q'R', we can calculate the lengths of its corresponding sides:
- Side P'Q': sqrt((3 - 4)^2 + (1 - 0)^2) = sqrt(1 + 1) = sqrt(2)
- Side Q'R': sqrt((-1 - 3)^2 + (-2 - 1)^2) = sqrt(16 + 9) = sqrt(25) = 5
- Side R'P': sqrt((-1 - 4)^2 + (-2 - 0)^2) = sqrt(25 + 4) = sqrt(29)
To find the scale factor, we can divide the corresponding side lengths of the two triangles:
- Scale factor = (Side P'Q' / Side PQ) = sqrt(2) / sqrt(8)
- Simplifying the square roots: sqrt(2) / sqrt(8) = sqrt(2/8) = sqrt(1/4) = 1/2
Therefore, the scale factor of the dilation that transforms triangle PQR to triangle P'Q'R' is 1/2. This means that each length in PQR is multiplied by 1/2 to get the corresponding length in P'Q'R'.
Part B: To reflect triangle P'Q'R' about the y-axis, we need to change the sign of the x-coordinates while leaving the y-coordinates unchanged.
The reflected coordinates of P' = (4, 0) will be (-4, 0)
The reflected coordinates of Q' = (3, 1) will be (-3, 1)
The reflected coordinates of R' = (-1, -2) will be (1, -2)
Therefore, the coordinates of triangle P"Q"R" obtained after P'Q'R' is reflected about the y-axis are P"(-4, 0), Q"(-3, 1), and R"(1, -2).
Part C: To determine if the two triangles PQR and P''Q''R'' are congruent, we need to compare their corresponding side lengths and angles.
Looking at the side lengths:
- Side PQ has a length of sqrt(8) in PQR, and in P''Q''R'' side length PQ" is also sqrt(8).
- Side QR has a length of 10 in PQR, and in P''Q''R'' side length Q"R" is also 10.
- Side RP has a length of sqrt(116) in PQR, and in P''Q''R'' side length R"P" is also sqrt(116).
All corresponding side lengths are equal, indicating that the triangles have the same size.
Moreover, reflecting a triangle about the y-axis does not change its shape, only its orientation. Therefore, the angles of the two triangles remain the same.
Since the triangles have the same side lengths and angles, we can conclude that triangles PQR and P''Q''R'' are congruent.