Question

△abc is rotated around point e by an angle t.
which statement is true about △abc and △abc?
the area of △abc is twice the area of △abc.
the corresponding sides of △abc and △abc are congruent.
the corresponding angles of △abc and △abc are different measures.
the lengths of the sides of △abc are equal to the lengths of the sides of △abc only when t = 90°.

Explanation:

Step1: Properties of rotation

Rotation is a rigid transformation. Rigid transformations (such as rotation) preserve the shape and size of a figure.

Step2: Analyzing each option

• Option 1:

Since rotation is a rigid transformation, the area of \(\triangle ABC\) is equal to the area of \(\triangle A'B'C'\). So, the statement "The area of \(\triangle ABC\) is twice the area of \(\triangle A'B'C'\)" is false.

• Option 2:

Because rotation is a rigid transformation, \(\triangle ABC\cong\triangle A'B'C'\). By the definition of congruent triangles, the corresponding sides of \(\triangle ABC\) and \(\triangle A'B'C'\) are congruent.

• Option 3:

Since \(\triangle ABC\cong\triangle A'B'C'\) (due to rotation being a rigid transformation), the corresponding angles of \(\triangle ABC\) and \(\triangle A'B'C'\) have the same measure. So, the statement "The corresponding angles of \(\triangle ABC\) and \(\triangle A'B'C'\) are different measures" is false.

• Option 4:

The lengths of the sides of \(\triangle ABC\) are equal to the lengths of the sides of \(\triangle A'B'C'\) for any value of \(t\) (because rotation is a rigid transformation). So, the statement "The lengths of the sides of \(\triangle ABC\) are equal to the lengths of the sides of \(\triangle A'B'C'\) only when \(t = 90^{\circ}\)" is false.

Answer:

The corresponding sides of \(\triangle ABC\) and \(\triangle A'B'C'\) are congruent.