Question

triangle $\triangle abc$ is the result of rotating $\triangle abc$ by $-135^{\circ}$ about point $a$.

graph of triangle $abc$ with point $a$ at $(-3, 1)$, point $b$ at $(-5, -6)$, point $c$ at $(-2, -2)$ on a coordinate plane

select all of the correct statements about the unchanged properties of $\triangle abc$ and $\triangle abc$.
choose all answers that apply:
a $a$ and $a$ have the same coordinates.
b the perimeters of $\triangle abc$ and $\triangle abc$ are the same.
c $\angle c$ and $\angle c$ have the same measures.
d none of the above

Explanation:

• Option A: When a figure is rotated about a point, the center of rotation (point \( A \)) remains fixed. So, \( A \) and \( A' \) have the same coordinates. This is correct.
• Option B: Rotation is a rigid transformation, which means the side lengths of the triangle remain unchanged. Since perimeter is the sum of side lengths, the perimeters of \( \triangle ABC \) and \( \triangle A'B'C' \) are the same. This is correct.
• Option C: Rotation is a rigid transformation, so the angle measures of the triangle remain unchanged. Thus, \( \angle C \) and \( \angle C' \) have the same measures. This is correct.
• Option D is incorrect because A, B, and C are correct.

Answer:

A. \( A \) and \( A' \) have the same coordinates.
B. The perimeters of \( \triangle ABC \) and \( \triangle A'B'C' \) are the same.
C. \( \angle C \) and \( \angle C' \) have the same measures.