Question

1. dilate quadrilateral wxyz with a scale factor of $\frac{1}{2}$ centered at (7,-2).

pw <1,-2>
px<8,4>
py<13,8>
pz<7,-1>
w( , )
x( , )
y( , )
z( , )

Explanation:

Step1: Recall dilation formula

For a point $(x,y)$ dilated by scale factor $k$ about center $(h,k_{center})$, the new point $(x',y')$ is:
$$x' = h + k(x - h)$$
$$y' = k_{center} + k(y - k_{center})$$
Here, $k=\frac{1}{2}$, center $(7,-2)$; pre-image points: $W(1,-2)$, $X(8,4)$, $Y(8,8)$, $Z(7,-1)$

Step2: Calculate $W'$

Substitute $W(1,-2)$:
$$x' = 7 + \frac{1}{2}(1-7) = 7 - 3 = 4$$
$$y' = -2 + \frac{1}{2}(-2+2) = -2 + 0 = -2$$

Step3: Calculate $X'$

Substitute $X(8,4)$:
$$x' = 7 + \frac{1}{2}(8-7) = 7 + 0.5 = 7.5$$
$$y' = -2 + \frac{1}{2}(4+2) = -2 + 3 = 1$$

Step4: Calculate $Y'$

Substitute $Y(8,8)$:
$$x' = 7 + \frac{1}{2}(8-7) = 7 + 0.5 = 7.5$$
$$y' = -2 + \frac{1}{2}(8+2) = -2 + 5 = 3$$

Step5: Calculate $Z'$

Substitute $Z(7,-1)$:
$$x' = 7 + \frac{1}{2}(7-7) = 7 + 0 = 7$$
$$y' = -2 + \frac{1}{2}(-1+2) = -2 + 0.5 = -1.5$$

Answer:

$W(4, -2)$
$X(7.5, 1)$
$Y(7.5, 3)$
$Z(7, -1.5)$