Question

1. rectangle pqrs with vertices p(5, 15), q(15, 15), r(15, 10), and s(5, 10): k = 4/5

p(_, _)
q(_, _)
r(_, _)
s(_, _)

1. square mnop with vertices m(-7, -4), n(-4, -3), o(-3, -6), and p(-6, -7): k = 2

m(_, _)
n(_, _)
o(_, _)
p(_, _)

Explanation:

Step1: Recall dilation formula

If a point $(x,y)$ is 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 the new coordinates for rectangle $PQRS$ with $k = \frac{4}{5}$

For point $P(5,15)$:
$x'_P=\frac{4}{5}\times5 = 4$ and $y'_P=\frac{4}{5}\times15 = 12$, so $P'(4,12)$.
For point $Q(15,15)$:
$x'_Q=\frac{4}{5}\times15 = 12$ and $y'_Q=\frac{4}{5}\times15 = 12$, so $Q'(12,12)$.
For point $R(15,10)$:
$x'_R=\frac{4}{5}\times15 = 12$ and $y'_R=\frac{4}{5}\times10 = 8$, so $R'(12,8)$.
For point $S(5,10)$:
$x'_S=\frac{4}{5}\times5 = 4$ and $y'_S=\frac{4}{5}\times10 = 8$, so $S'(4,8)$.

Step3: Find the new coordinates for square $MNOP$ with $k = 2$

For point $M(-7,-4)$:
$x'_M=2\times(-7)=-14$ and $y'_M=2\times(-4)=-8$, so $M'(-14,-8)$.
For point $N(-4,-3)$:
$x'_N=2\times(-4)=-8$ and $y'_N=2\times(-3)=-6$, so $N'(-8,-6)$.
For point $O(-3,-6)$:
$x'_O=2\times(-3)=-6$ and $y'_O=2\times(-6)=-12$, so $O'(-6,-12)$.
For point $P(-6,-7)$:
$x'_P=2\times(-6)=-12$ and $y'_P=2\times(-7)=-14$, so $P'(-12,-14)$.

Answer:

$P'(4,12)$
$Q'(12,12)$
$R'(12,8)$
$S'(4,8)$
$M'(-14,-8)$
$N'(-8,-6)$
$O'(-6,-12)$
$P'(-12,-14)$