Question

△rst is dilated by a scale factor of 5 with a center of dilation at the origin to form △rst. then △rst is dilated by a scale factor of $\frac{2}{5}$ with a center of dilation at the origin to form △rst. which statement is true? t has coordinates (2, -4). r has coordinates (15, -10). s has coordinates (-20, -15). r has coordinates (-4, 6).

Explanation:

Step1: Recall dilation formula

If a point $(x,y)$ is dilated by a scale - factor $k$ with the center of dilation at the origin, the new coordinates are $(kx,ky)$.
Let's assume the coordinates of the vertices of $\triangle RST$ are $R(x_R,y_R)$, $S(x_S,y_S)$, and $T(x_T,y_T)$. After the first dilation by a scale - factor of $k_1 = 5$, the coordinates of $\triangle R'S'T'$ are $R'(5x_R,5y_R)$, $S'(5x_S,5y_S)$, and $T'(5x_T,5y_T)$. Then, after the second dilation by a scale - factor of $k_2=\frac{2}{5}$, the coordinates of $\triangle R''S''T''$ are $R''(5x_R\times\frac{2}{5},5y_R\times\frac{2}{5})=(2x_R,2y_R)$, $S''(2x_S,2y_S)$, and $T''(2x_T,2y_T)$.

Step2: Assume initial coordinates (by observing the grid)

Let's assume $R(- 3,2)$, $S(-3,-3)$, $T(1,-2)$.
After the first dilation by a scale - factor of $5$:
$R'=(-3\times5,2\times5)=(-15,10)$
$S'=(-3\times5,-3\times5)=(-15,-15)$
$T'=(1\times5,-2\times5)=(5,-10)$
After the second dilation by a scale - factor of $\frac{2}{5}$:
$R''=(-15\times\frac{2}{5},10\times\frac{2}{5})=(-6,4)$
$S''=(-15\times\frac{2}{5},-15\times\frac{2}{5})=(-6,-6)$
$T''=(5\times\frac{2}{5},-10\times\frac{2}{5})=(2,-4)$

Answer:

$T'$ has coordinates $(2, - 4)$.