Question

1. triangle nop below will be dilated with the origin as the center of dilation and a scale factor of 1/2. what will be the coordinates of the vertices of the dilated image, △nop? a. n(4,4), o(8,4), p(16,20) b. n(1,2), o(2,2), p(4,10) c. n(1,1), o(2,1), p(4,5) d. n(1,1), o(1,1), p(3,5) 4. triangle rst below will be dilated with the origin as the center of dilation and a scale factor of 4. what will be the coordinates of the vertices of the dilated image, △rst? a. r(4,12), s(12,12), t(8,4) b. r(4,12), s(9,9), t(8,1) c. r(2,6), s(12,12), t(4,2) d. r(1/4,3/4), s(1/2,1/4), t(1,5/4)

Explanation:

Step1: Recall dilation formula

If a point $(x,y)$ is dilated with the origin $(0,0)$ as the center of dilation and a scale - factor $k$, the new coordinates $(x',y')$ are given by $(x',y')=(k\cdot x,k\cdot y)$.

Step2: Analyze first triangle NOP

Let's assume the coordinates of $\triangle NOP$ are $N(x_1,y_1)$, $O(x_2,y_2)$, $P(x_3,y_3)$ and the scale - factor $k = \frac{1}{2}$.
For a point $(x,y)$ dilated about the origin with scale - factor $k=\frac{1}{2}$, the new coordinates are $(\frac{1}{2}x,\frac{1}{2}y)$.

Step3: Analyze second triangle RST

For $\triangle RST$ dilated with a scale - factor $k = 4$ about the origin, if a point $(x,y)$ in $\triangle RST$ has coordinates, the new coordinates $(x',y')$ of the dilated point are $(4x,4y)$.

For the first triangle $\triangle NOP$ with scale - factor $\frac{1}{2}$:
If $N(4,4)$, $O(2,2)$, $P(8,4)$, then $N'(4\times\frac{1}{2},4\times\frac{1}{2})=(2,2)$, $O'(2\times\frac{1}{2},2\times\frac{1}{2})=(1,1)$, $P'(8\times\frac{1}{2},4\times\frac{1}{2})=(4,2)$.
For the second triangle $\triangle RST$ with scale - factor $4$:
If $R(1,3)$, $S(3,3)$, $T(2,1)$, then $R'(1\times4,3\times4)=(4,12)$, $S'(3\times4,3\times4)=(12,12)$, $T'(2\times4,1\times4)=(8,4)$.

Answer:

For the dilation of $\triangle NOP$ with scale - factor $\frac{1}{2}$ about the origin, the new coordinates are $N'(2,2)$, $O'(1,1)$, $P'(4,2)$. For the dilation of $\triangle RST$ with scale - factor $4$ about the origin, the new coordinates are $R'(4,12)$, $S'(12,12)$, $T'(8,4)$. But since this is a multiple - choice problem, for the dilation of $\triangle RST$ with scale - factor $4$ about the origin:
If $R(1,3)$, $S(3,3)$, $T(2,1)$, then $R'(4\times1,4\times3)=(4,12)$, $S'(4\times3,4\times3)=(12,12)$, $T'(4\times2,4\times1)=(8,4)$. The answer for the dilation of $\triangle RST$ is A. $R'(4,12),S'(12,12),T'(8,4)$