Question

lar figures
triangle def was dilated according to the rule $d_{o, \frac{1}{3}} (x, y) \to (\frac{1}{3}x, \frac{1}{3}y)$ to create similar triangle def.
which statements are true? choose three correct answers.
$\triangle def \sim \triangle def$
the distance from point d to the origin is $\frac{1}{3}$ the distance of point d to the origin.
$\angle f$ corresponds to $\angle f$.
the measure of $\angle e$ is $\frac{1}{3}$ the measure of $\angle e$.

Explanation:

Step1: Analyze Similarity from Dilation

Dilation is a similarity transformation, so dilated figures are similar. Thus, $\triangle DEF \sim \triangle D'E'F'$ is true.

Step2: Analyze Distance from Origin

The dilation rule is $D_{O,\frac{1}{3}}(x,y)\to(\frac{1}{3}x,\frac{1}{3}y)$, so coordinates of $D'$ are $\frac{1}{3}$ of $D$'s. By distance formula, distance from origin to $D'$ is $\frac{1}{3}$ of that to $D$. So this statement is true.

Step3: Analyze Corresponding Angles

In similar triangles, corresponding angles are equal and correspond. $\angle F$ and $\angle F'$ are corresponding angles, so this is true.

Step4: Analyze Angle Measure

In similar triangles, corresponding angles are congruent (equal measure), not scaled. So "The measure of $\angle E'$ is $\frac{1}{3}$ the measure of $\angle E$" is false.

Answer:

• $\triangle DEF \sim \triangle D'E'F'$
• The distance from point $D'$ to the origin is $\frac{1}{3}$ the distance of point $D$ to the origin.
• $\angle F$ corresponds to $\angle F'$.