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)
Ok, after you plotted the two triangles, did you not notice that ...
PartA: somebody simply took half of the coordinates of the first
triangle to get the 2nd triangle ?
So what is the scale factor?
PartB: if any point is reflected in the y-axis, its y value stays the same, but
but its x value becomes the opposite.
so Q'(3,1) ----> Q''(-3,1) etc
PartC: to be congruent, the two triangles must have all corresponding sides and angles equal to each other. Is that the case?
Do they even have the same area??
To plot the triangles, we will use a coordinate grid.
First, let's plot the original triangle PQR:
P(8, 0)
Q(6, 2)
R(-2, -4)
Next, let's plot the transformed triangle P'Q'R':
P'(4, 0)
Q'(3, 1)
R'(-1, -2)
Here is the coordinate grid with the plotted triangles:
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-10 0 10
Part A: The scale factor of the dilation that transforms triangle PQR to triangle P'Q'R' can be found by comparing the corresponding side lengths. Let's consider the side PQ: the length of PQ is √[(6-8)^2 + (2-0)^2] = √(4^2 + 2^2) = √(16 + 4) = √20. The length of P'Q' is √[(3-4)^2 + (1-0)^2] = √(1^2 + 1^2) = √2. Therefore, the scale factor is √2 / √20 = 1/√10, which simplifies to √10 / 10. The same scale factor applies to all corresponding side lengths, so the scale factor of the dilation is √10 / 10.
Part B: To reflect triangle P'Q'R' about the y-axis, we simply change the sign of the x-coordinates. Therefore, the coordinates of the reflected triangle P''Q''R'' are:
P''(-4, 0)
Q''(-3, 1)
R''(1, -2)
Part C: The triangles PQR and P''Q''R'' are not congruent. Congruent triangles have the same shape and size, but PQR and P''Q''R'' have different side lengths and angles. However, triangles PQR and P'Q'R' are similar because they have the same shape, just different sizes. Their corresponding angles are congruent, and their corresponding side lengths are proportional with a scale factor of √10 / 10.
To answer these questions, let's start by plotting triangles PQR and P'Q'R' on a coordinate grid.
Plotting Triangle PQR:
P(8, 0)
Q(6, 2)
R(-2, -4)
Plotting Triangle P'Q'R':
P'(4, 0)
Q'(3, 1)
R'(-1, -2)
Now, let's move on to answering each part of the question:
Part A: What is the scale factor of the dilation that transforms triangle PQR to triangle P'Q'R'?
To determine the scale factor, we need to compare the corresponding side lengths of the two triangles. Let's consider side PQ in triangle PQR and side P'Q' in triangle P'Q'R'.
Distance PQ:
√[(6-8)^2 + (2-0)^2] = √[4 + 4] = √8
Distance P'Q':
√[(3-4)^2 + (1-0)^2] = √[1 + 1] = √2
To find the scale factor, we divide the length of P'Q' by the length of PQ.
Scale factor = √2 / √8 = (√2 / √(2 * 4)) = (√2 / (2 * √2)) = 1 / 2
So, the scale factor of the dilation that transforms triangle PQR to triangle P'Q'R' is 1/2.
Part B: Write the coordinates of triangle P''Q''R'' obtained after P'Q'R' is reflected about the y-axis.
When we reflect a point about the y-axis, we need to change the sign of its x-coordinate.
P''(−4, 0)
Q''(−3, 1)
R''(1, -2)
So, 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: Are the two triangles PQR and P''Q''R'' congruent?
To determine if two triangles are congruent, we need to compare their corresponding sides and angles.
Looking at triangles PQR and P''Q''R'':
- Side lengths PQ and P''Q'' are equal.
- Side lengths QR and Q''R'' are equal.
- Side lengths RP and R''P'' are equal.
Therefore, the three pairs of corresponding side lengths are equal, and the triangles have the same shape and size.
Also, since we performed a reflection, the angles between the sides are preserved.
Based on this, we can conclude that triangles PQR and P''Q''R'' are congruent.
I hope this explanation helps you understand the process of solving the problem step by step.