Question

tanya performs two transformations on △abc to form △abc, as shown on the coordinate grid below. determine whether each statement about the triangles is true. choose true or false from each drop - down menu for each statement. △abc is similar to △abc. △abc is congruent to △abc. △abc can be reflected across the y - axis to form △abc, and then △abc can be rotated 90° clockwise about the origin to form △abc. △abc can be rotated 90° counterclockwise about the origin to form △abc, and then △abc can be dilated by a scale factor of 2 with the center of dilation at the origin to form △abc.

Explanation:

Step1: Recall similarity and congruence rules

Similar triangles have same - shape (equal angles), and congruent triangles have same - shape and size. Rigid motions (translations, rotations, reflections) preserve congruence.

Step2: Analyze similarity

Since rigid motions (the two transformations) preserve shape, $\triangle ABC$ and $\triangle A''B''C''$ are similar. So, $\triangle ABC$ is similar to $\triangle A''B''C''$ is True.

Step3: Analyze congruence

Rigid motions preserve size and shape. Since the two transformations are rigid motions, $\triangle ABC$ is congruent to $\triangle A''B''C''$ is True.

Step4: Check transformation sequence 1

If we reflect $\triangle ABC$ across the $y$ - axis, the $x$ - coordinates of the vertices change sign. Then rotating $90^{\circ}$ clockwise about the origin $(x,y)\to(y, - x)$. By checking the coordinates, this sequence is False.

Step5: Check transformation sequence 2

Rotating $90^{\circ}$ counter - clockwise about the origin $(x,y)\to(-y,x)$. Dilating by a scale factor of 2 with the center of dilation at the origin $(x,y)\to(2x,2y)$. By checking the coordinates of the vertices, this sequence is False.

Answer:

1. True
2. True
3. False
4. False