Question

triangle rst with vertices r(-5, 1), s(-3, 4), and t(2, -1): k = 2
rectangle abcd with vertices a(-3, 0), b(1, 2), c(2, 0), and d(-2, -2): k = 3
dilations

Explanation:

Step1: Recall dilation formula

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

Step2: Find dilated vertices of triangle RST

Given $R(-5,1)$, $S(-3,4)$, $T(2,-1)$ and $k = 2$.
For point $R$: $R'=(2\times(-5),2\times1)=(-10,2)$.
For point $S$: $S'=(2\times(-3),2\times4)=(-6,8)$.
For point $T$: $T'=(2\times2,2\times(-1))=(4,-2)$.

Step3: Find dilated vertices of rectangle ABCD

Given $A(-3,0)$, $B(1,2)$, $C(2,0)$, $D(-2,-2)$ and $k = 3$.
For point $A$: $A'=(3\times(-3),3\times0)=(-9,0)$.
For point $B$: $B'=(3\times1,3\times2)=(3,6)$.
For point $C$: $C'=(3\times2,3\times0)=(6,0)$.
For point $D$: $D'=(3\times(-2),3\times(-2))=(-6,-6)$.

Answer:

Vertices of dilated triangle R'S'T': $R'(-10,2)$, $S'(-6,8)$, $T'(4,-2)$; Vertices of dilated rectangle A'B'C'D': $A'(-9,0)$, $B'(3,6)$, $C'(6,0)$, $D'(-6,-6)$