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. (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)
A: Use the distance formula to find lengths PQ and P'Q'
The scale factor is P'Q' / PQ
B: (x,y) → (-x,y)
C: Unless the scale factor from (A) is 1, the two are not congruent, since reflection does not change any side lengths.
oops. For (A), you need to check all three sides. to be sure the ratios are all the same. It might be easier just to make sure that the coordinates have a common ratio.
To plot the triangles PQR and P'Q'R', we will use a coordinate grid.
The given coordinates are:
P(3, -6), Q(0, 9), R(-3, 0)
P'(1, -2), Q'(0, 3), R'(-1, 0)
Using these coordinates, we can plot the triangles on a coordinate grid.
Part A:
To find the scale factor of the dilation that transforms triangle PQR to triangle P'Q'R', we can compare the distances between corresponding points.
The distance between P'(1, -2) and P(3, -6) is:
√[(1-3)^2 + (-2-(-6))^2] = √[4 + 16] = √20 = 2√5
The distance between Q'(0, 3) and Q(0, 9) is:
√[(0-0)^2 + (3-9)^2] = √[0 + 36] = √36 = 6
The distance between R'(-1, 0) and R(-3, 0) is:
√[(-1-(-3))^2 + (0-0)^2] = √[4 + 0] = √4 = 2
The scale factor is the ratio of the corresponding distances:
Scale factor = (Distance in P'Q'R') / (Distance in PQR)
Therefore, the scale factor of the dilation is:
2√5 / 6 = √5 / 3
The scale factor is √5 / 3.
Part B:
To reflect triangle P'Q'R' about the y-axis, we need to change the sign of the x-coordinates.
The coordinates of the reflected triangle, P"Q"R", are:
P"(-1, -2), Q"(0, 3), R"(1, 0)
Part C:
To determine if triangles PQR and P''Q''R'' are congruent, we compare their corresponding side lengths and angles.
Comparing the side lengths:
PQ = P"Q" = √[(0-3)^2 + (9-(-6))^2] = √[9 + 225] = √234
QR = Q"R" = √[(0-(-3))^2 + (9-0)^2] = √[9 + 81] = √90
RP = R"P" = √[(-3-1)^2 + (0-(-2))^2] = √[16 + 4] = √20 = 2√5
Comparing the angles:
∠P = ∠P", ∠Q = ∠Q", ∠R = ∠R"
Since the side lengths and angles of triangles PQR and P''Q''R'' are equal, the two triangles are congruent.
Before we proceed to answer the questions, let's plot the given triangles on a coordinate grid to visualize them.
Triangle PQR:
P(3, -6), Q(0, 9), R(-3, 0)
Triangle P'Q'R':
P'(1, -2), Q'(0, 3), R'(-1, 0)
Here is a visual representation of the two triangles:
Triangle PQR:
P(3, -6) -- Q(0, 9)
\ /
\ /
R(-3, 0)
Triangle P'Q'R':
P'(1, -2) -- Q'(0, 3)
\ /
\ /
R'(-1, 0)
Now, let's proceed to answer 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 a dilation, we can compare the lengths of corresponding sides of the two triangles.
For example, let's consider side PQ in triangle PQR and side P'Q' in triangle P'Q'R':
Length of PQ = sqrt((0 - 3)^2 + (9 - (-6))^2) = sqrt(3^2 + 15^2) = sqrt(9 + 225) = sqrt(234)
Length of P'Q' = sqrt((0 - 1)^2 + (3 - (-2))^2) = sqrt(1^2 + 5^2) = sqrt(1 + 25) = sqrt(26)
The scale factor is the ratio of the corresponding side lengths:
Scale factor = Length of P'Q' / Length of PQ
= sqrt(26) / sqrt(234)
To simplify this, we can multiply both the numerator and denominator by sqrt(234) to eliminate the square roots in the denominator:
Scale factor = (sqrt(26) * sqrt(234)) / (sqrt(234) * sqrt(234))
= sqrt(6084) / 234
= 78 / 234
= 1 / 3
Therefore, the scale factor of the dilation that transforms triangle PQR to triangle P'Q'R' is 1/3.
Part B: Write the coordinates of triangle P"Q"R" obtained after P'Q'R' is reflected about the y-axis.
To reflect a point about the y-axis, we only need to negate the x-coordinate. Reflecting each point of triangle P'Q'R' in this way, we obtain triangle P"Q"R":
P"(-1, -2), Q"(0, 3), R"(1, 0)
So, the coordinates of triangle P"Q"R" after reflection about the y-axis are 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.
Two triangles are congruent if their corresponding sides and angles are equal. Let's check if the corresponding sides and angles of triangles PQR and P"Q"R" are equal:
Sides:
- Side PQ and side P"Q" have equal lengths.
- Side QR and side Q"R" have equal lengths.
- Side RP and side R"P" have equal lengths.
Angles:
- Angle PQR and angle P"Q"R" have equal measures.
- Angle QRP and angle Q"R"P" have equal measures.
- Angle RPQ and angle R"P"Q" have equal measures.
Since all corresponding sides and angles of triangles PQR and P"Q"R" are equal, we can conclude that these triangles are congruent.
Therefore, triangles PQR and P"Q"R" are congruent.