Question

1. rotate trapezoid mnop with m(3, -2), n(6, -2), o(8, -5) and p(1, -5) 180° about the origin. what are the coordinates of o?
2. rotate triangle pqr with p(-1, -3), q(-5, -7) and r(-7, -3) 270° counterclockwise about the origin. what are the coordinates of p?
3. dilate triangle klm with k(-2, 1), l(-3, -4) and m(-4, 1) with using a scale factor of 2. what are the coordinates of k?
4. dilate rectangle wxyz with w(3, 9), x(9, 9), y(9, 6) and z(3, 6) using a scale factor of 1/3. what are the coordinates of y?

Explanation:

Step1: Recall rotation rule for 180 - degree rotation

The rule for rotating a point $(x,y)$ 180 - degrees about the origin is $(x,y)\to(-x,-y)$.
For point $O(8, - 5)$, applying the rule:
$x = 8$ and $y=-5$.
New $x=-8$ and new $y = 5$. So the coordinates of $O'$ are $(-8,5)$.

Step2: Recall rotation rule for 270 - degree counter - clockwise rotation

The rule for rotating a point $(x,y)$ 270 - degrees counter - clockwise about the origin is $(x,y)\to(y,-x)$.
For point $P(-1,-3)$, applying the rule:
$x=-1$ and $y = - 3$.
New $x=-3$ and new $y = 1$. So the coordinates of $P'$ are $(-3,1)$.

Step3: Recall dilation rule

The rule for dilating a point $(x,y)$ by a scale factor $k$ is $(x,y)\to(kx,ky)$.
For point $K(-2,1)$ with $k = 2$:
$x=-2$ and $y = 1$.
New $x=2\times(-2)=-4$ and new $y=2\times1 = 2$. So the coordinates of $K'$ are $(-4,2)$.

Step4: Recall dilation rule for scale factor $\frac{1}{3}$

The rule for dilating a point $(x,y)$ by a scale factor $k=\frac{1}{3}$ is $(x,y)\to(\frac{1}{3}x,\frac{1}{3}y)$.
For point $Y(9,6)$:
$x = 9$ and $y = 6$.
New $x=\frac{1}{3}\times9 = 3$ and new $y=\frac{1}{3}\times6=2$. So the coordinates of $Y'$ are $(3,2)$.

Answer:

1. $(-8,5)$
2. $(-3,1)$
3. $(-4,2)$
4. $(3,2)$