Question

WILL MARK BRAINLIST!! 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.

Answer 1

Answer:

The scale factor of the dillation that transforms triangle PQR to triangle P'Q'R' is $\frac{1}{3}$.

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:

$(x',y') = O(x,y)+k\cdot (x,y)$ (Eq. 1)

Where:

$O(x,y)$ - Origin, dimensionless.

$k$ - Scale factor, dimensionless.

$(x,y)$ - Original point, dimensionless.

$(x',y')$ - Dilated point, dimensionless.

If we consider that $O(x,y) = (0,0)$ and we know that $P(x,y) =(3,-6)$, $Q(x,y) =(0,-9)$, $R(x,y) =(-3,0)$, $P'(x,y) =(1,-2)$, $Q'(x,y) =(0,-3)$ and $R'(x,y) = (-1,0)$, then scale factor of each point is, respectively:

Point P

$P'(x,y) = O(x,y)+k\cdot P(x,y)$

$(1,-2) = (0,0)+k\cdot (3,-6)$

$(1,-2) = (3\cdot k, -6\cdot k)$

$k = \frac{1}{3}$

Point Q

$Q'(x,y) = O(x,y)+k\cdot Q(x,y)$

$(0,-3) = (0,0)+k\cdot (0,-9)$

$(0,-3) = (0,-9\cdot k)$

$k = \frac{1}{3}$

Point R

$R'(x,y) = O(x,y)+k\cdot R(x,y)$

$(-1,0) = (0,0)+k\cdot (-3,0)$

$(-1, 0) = (-3\cdot k, 0)$

$k = \frac{1}{3}$

Hence, the scale factor of the dillation that transforms triangle PQR to triangle P'Q'R' is $\frac{1}{3}$.