Question

1. triangle xyz has been enlarged with p as the center of dilation to form △xyz. identify the equivalent ratios that are equal to the scale factor.
2. a triangle is dilated with center of dilation at point u. point e is a vertex of the triangle, and point e is the corresponding vertex of the image. if ue = 2 centimeters and ue = 10 centimeters, what is the scale factor?
3. the coordinates of quadrilateral abcd are a(-6, 2), b(-5, 3), c(7, 3), and d(0, -4). what are the coordinates of the image if the quadrilateral is translated 4 units right and 3 units down? a(-2,-1) b(-1,0) (x + 4,y - 3)
4. the coordinates of △jkl are j(0, 1), k(6, 0), and l(-6, 0). what are the coordinates of the image if the triangle is translated 8 units left?

Explanation:

Step1: Recall scale - factor formula

The scale factor $k$ of a dilation is given by the ratio of the distance from the center of dilation to a point on the image to the distance from the center of dilation to the corresponding point on the pre - image, i.e., $k=\frac{\text{distance to image}}{\text{distance to pre - image}}$.

Step2: Solve for scale factor in problem 2

Given $UE = 2$ cm and $UE'=10$ cm. Using the formula $k=\frac{UE'}{UE}$, we substitute the values: $k=\frac{10}{2}=5$.

Step3: Recall translation rule

For a translation $h$ units right and $v$ units down, the rule for a point $(x,y)$ is $(x + h,y - v)$. Here $h = 4$ and $v=3$.

Step4: Find new coordinates of quadrilateral in problem 3

For point $A(-6,2)$: $A'=(-6 + 4,2-3)=(-2,-1)$.
For point $B(-5,3)$: $B'=(-5 + 4,3 - 3)=(-1,0)$.
For point $C(7,3)$: $C'=(7 + 4,3-3)=(11,0)$.
For point $D(0,-4)$: $D'=(0 + 4,-4-3)=(4,-7)$.

Step5: Apply translation rule to triangle in problem 4

For a translation 8 units left ($h=-8$ and $v = 0$), the rule for a point $(x,y)$ is $(x+h,y)$.
For point $J(0,1)$: $J'=(0-8,1)=(-8,1)$.
For point $K(6,0)$: $K'=(6 - 8,0)=(-2,0)$.
For point $L(-6,0)$: $L'=(-6-8,0)=(-14,0)$.

Answer:

1. The scale factor is 5.
2. $A'(-2,-1)$, $B'(-1,0)$, $C'(11,0)$, $D'(4,-7)$
3. $J'(-8,1)$, $K'(-2,0)$, $L'(-14,0)$