Question

which of the following statements is not true about $\triangle abc$ and its image $\triangle abc$ under a rotation?
\bigcirc the area of $\triangle abc$ is the same as $\triangle abc$.
\bigcirc the interior angles of $\triangle abc$ have twice the measure of the interior angles of $\triangle abc$.
\bigcirc the perimeter of $\triangle abc$ is the same as $\triangle abc$.
\bigcirc the distance from point $a$ to point $c$ is the same as the distance from point $a$ to point $c$.
\boxed{calculator}

Explanation:

Step1: Recall Rotation Properties

A rotation is a rigid transformation, meaning it preserves the shape and size of the figure. So, corresponding sides and angles remain equal.

Step2: Analyze Each Option

• Area: Rigid transformations preserve area, so area of \( \triangle ABC \) and \( \triangle A'B'C' \) is same. True.
• Angles: Rigid transformations preserve angle measures. So interior angles of \( \triangle ABC \) and \( \triangle A'B'C' \) are equal, not twice. False.
• Perimeter: Since side lengths are preserved, perimeter (sum of sides) is same. True.
• Distance \( A \) to \( C \): Corresponding sides (like \( AC \) and \( A'C' \)) are equal, so distance is same. True.

Answer:

The interior angles of \( \triangle ABC \) have twice the measure of the interior angles of \( \triangle A'B'C' \).