Answer:
The scale factor of the dillation that transforms triangle PQR to triangle P'Q'R' is [tex]\frac{1}{3}[/tex].
Step-by-step explanation:
There is a typographical mistake in the statement. Correct statement is:
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). What is the scale factor of the dilation that transforms triangle PQR to triangle P'Q'R? Explain your answer.
From Linear Algebra we define dillation with respect to a given point as:
[tex](x',y') = O(x,y)+k\cdot (x,y)[/tex] (Eq. 1)
Where:
[tex]O(x,y)[/tex] - Origin, dimensionless.
[tex]k[/tex] - Scale factor, dimensionless.
[tex](x,y)[/tex] - Original point, dimensionless.
[tex](x',y')[/tex] - Dilated point, dimensionless.
If we consider that [tex]O(x,y) = (0,0)[/tex] and we know that [tex]P(x,y) =(3,-6)[/tex], [tex]Q(x,y) =(0,-9)[/tex], [tex]R(x,y) =(-3,0)[/tex], [tex]P'(x,y) =(1,-2)[/tex], [tex]Q'(x,y) =(0,-3)[/tex] and [tex]R'(x,y) = (-1,0)[/tex], then scale factor of each point is, respectively:
Point P
[tex]P'(x,y) = O(x,y)+k\cdot P(x,y)[/tex]
[tex](1,-2) = (0,0)+k\cdot (3,-6)[/tex]
[tex](1,-2) = (3\cdot k, -6\cdot k)[/tex]
[tex]k = \frac{1}{3}[/tex]
Point Q
[tex]Q'(x,y) = O(x,y)+k\cdot Q(x,y)[/tex]
[tex](0,-3) = (0,0)+k\cdot (0,-9)[/tex]
[tex](0,-3) = (0,-9\cdot k)[/tex]
[tex]k = \frac{1}{3}[/tex]
Point R
[tex]R'(x,y) = O(x,y)+k\cdot R(x,y)[/tex]
[tex](-1,0) = (0,0)+k\cdot (-3,0)[/tex]
[tex](-1, 0) = (-3\cdot k, 0)[/tex]
[tex]k = \frac{1}{3}[/tex]
Hence, the scale factor of the dillation that transforms triangle PQR to triangle P'Q'R' is [tex]\frac{1}{3}[/tex].
