All side lengths in quadrilateral PQRS measure 4 units. A transformation maps PQRS to P'Q'R'S'. All sides of P'Q'R'S' measure 1
2 answers:
Answer:
No, the side lengths are not preserved.
Step by step explanation:
We have been given that all side lengths in quadrilateral PQRS measure 4 units. A transformation maps PQRS to P'Q'R'S'. All sides of P'Q'R'S' measure 1 unit.
We can see that the transformation that maps PQRS to P'Q'R'S' is dilation as after dilating our pre-image PQRS with a factor of we get our image P'Q'R'S'.
Since we know that isometry does not change the shape or size of the figure and isometric figures are said to be geometrically congruent. In our case our side lengths are shrunken so the given transformation is not isometric.
Therefore, our transformation is not isometric as the side lengths are not preserved.
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Answer:
Please find attached the required plot accomplished with an online tool
Part A:
1/4
Part B:
P''(-1, 0), Q''(0, -1), and R''(2, -1)
Part C:
Triangle PQR is similar to triangle P''Q''R'' but they are not congruent
Step-by-step explanation:
Part A:
Triangle ΔPQR has vertices P(4, 0), Q(0, -4), R(-8, -4)
Triangle ΔP'Q'R' has vertices P'(1, 0), Q'(0, -1), R'(-2, -1)
The dimensions of the sides of the triangle are given by the relation;
Where;
(x₁, y₁) and (x₂, y₂) are the coordinates on the ends of the segment
For segment PQ, we place (x₁, y₁) = (4, 0) and (x₂, y₂) = (0, -4);
By substitution into the length equation, we get;
The length of segment PQ = 4·√2
The length of segment PR = 4·√10
The length of segment RQ = 8
The length of segment P'Q' = √2
The length of segment P'R' = √10
The length of segment R'Q' = 2
Therefore, the scale factor of the dilation of ΔPQR to ΔP'Q'R' is 1/4
Part B:
Reflection of (x, y) across the y-axis gives;
(x, y) image after reflection across the y-axis = (-x, y)
The coordinates after reflection of P'(1, 0), Q'(0, -1), R'(-2, -1) across the y-axis is given as follows;
P'(1, 0) image after reflection across the y-axis = P''(-1, 0)
Q'(0, -1) image after reflection across the y-axis = Q''(0, -1)
R'(-2, -1) image after reflection across the y-axis = R''(2, -1)
Part C:
Triangle PQR is similar to triangle P''Q''R'' but they are not congruent as the dimensions of the sides of triangle PQR and P''Q''R'' are not the same.
