Question

graph the image of $\triangle def$ after a dilation with a scale factor of $\frac{1}{2}$, centered at the origin.

Explanation:

Step1: Identify original coordinates

From the graph:
$D(10, -4)$, $E(10, 10)$, $F(6, -8)$

Step2: Apply dilation formula

For a dilation centered at the origin with scale factor $k=\frac{1}{2}$, the new coordinates $(x', y')$ are given by $(x', y')=(k \cdot x, k \cdot y)$

• For $D$: $x'=\frac{1}{2} \cdot 10=5$, $y'=\frac{1}{2} \cdot (-4)=-2$ → $D'(5, -2)$
• For $E$: $x'=\frac{1}{2} \cdot 10=5$, $y'=\frac{1}{2} \cdot 10=5$ → $E'(5, 5)$
• For $F$: $x'=\frac{1}{2} \cdot 6=3$, $y'=\frac{1}{2} \cdot (-8)=-4$ → $F'(3, -4)$

Step3: Plot and connect new points

Plot $D'(5, -2)$, $E'(5, 5)$, $F'(3, -4)$ on the grid, then connect the points to form $\triangle D'E'F'$.

Answer:

The coordinates of the dilated triangle are $D'(5, -2)$, $E'(5, 5)$, $F'(3, -4)$. When plotted and connected, this forms the image of $\triangle DEF$ after the dilation.