To solve these problems, we need to find the coordinates of the original points and then multiply them by the given
scale factor to find the new "dilated" coordinates.
The rule for dilation centered at the origin $(0,0)$ is:
$$ (x, y) \rightarrow (\text{scale factor} \cdot x, \text{scale factor} \cdot y) $$
Let's go through each problem step-by-step.
Problem 1
Scale Factor = 2
First, identify the coordinates of the original triangle $PQR$ from the graph:
* Point $P$ is at $(-1, -1)$.
* Point $Q$ is at $(2, -1)$.
* Point $R$ is at $(-1, 2)$.
Now, multiply each coordinate by the scale factor of
2:
*
$P'$: $(-1 \cdot 2, -1 \cdot 2) = (-2, -2)$
*
$Q'$: $(2 \cdot 2, -1 \cdot 2) = (4, -2)$
*
$R'$: $(-1 \cdot 2, 2 \cdot 2) = (-2, 4)$
Problem 2
Scale Factor = 0.5
First, identify the coordinates of the original line segment $XY$ from the graph:
* Point $X$ is at $(2, 2)$.
* Point $Y$ is at $(8, 8)$.
Now, multiply each coordinate by the scale factor of
0.5 (which is the same as dividing by 2):
*
$X'$: $(2 \cdot 0.5, 2 \cdot 0.5) = (1, 1)$
*
$Y'$: $(8 \cdot 0.5, 8 \cdot 0.5) = (4, 4)$
Problem 3
Scale Factor = $\frac{1}{2}$
First, identify the coordinates of the original diamond $KLMN$ from the graph:
* Point $K$ is at $(-2, 0)$.
* Point $L$ is at $(0, -2)$.
* Point $M$ is at $(2, 0)$.
* Point $N$ is at $(0, 2)$.
Now, multiply each coordinate by the scale factor of
$\frac{1}{2}$ (divide by 2):
*
$K'$: $(-2 \cdot \frac{1}{2}, 0 \cdot \frac{1}{2}) = (-1, 0)$
*
$L'$: $(0 \cdot \frac{1}{2}, -2 \cdot \frac{1}{2}) = (0, -1)$
*
$M'$: $(2 \cdot \frac{1}{2}, 0 \cdot \frac{1}{2}) = (1, 0)$
*
$N'$: $(0 \cdot \frac{1}{2}, 2 \cdot \frac{1}{2}) = (0, 1)$
Problem 4
Scale Factor = 3
First, identify the coordinates of the original diamond $KLMN$ from the graph:
* Point $K$ is at $(-1, 0)$.
* Point $L$ is at $(0, -1)$.
* Point $M$ is at $(1, 0)$.
* Point $N$ is at $(0, 1)$.
Now, multiply each coordinate by the scale factor of
3:
*
$K'$: $(-1 \cdot 3, 0 \cdot 3) = (-3, 0)$
*
$L'$: $(0 \cdot 3, -1 \cdot 3) = (0, -3)$
*
$M'$: $(1 \cdot 3, 0 \cdot 3) = (3, 0)$
*
$N'$: $(0 \cdot 3, 1 \cdot 3) = (0, 3)$
──────────────────────────────────────
Final Answer:
1) Scale Factor = 2
$P': (-2, -2)$
$Q': (4, -2)$
$R': (-2, 4)$
2) Scale Factor = 0.5
$X': (1, 1)$
$Y': (4, 4)$
3) Scale Factor = $\frac{1}{2}$
$K': (-1, 0)$
$L': (0, -1)$
$M': (1, 0)$
$N': (0, 1)$
4) Scale Factor = 3
$K': (-3, 0)$
$L': (0, -3)$
$M': (3, 0)$
$N': (0, 3)$
Parent Tip: Review the logic above to help your child master the concept of dilation worksheet middle school.
