Question

Two right triangles of equal dimensions in different orientations. Triangle upper G upper H upper J is on the left and triangle upper Z upper N upper P is on the right.
To prove the triangles are congruent, which of the following rigid motions would map △GHJ
to △ZNP
?

(1 point)
Responses

rotation −90°
(clockwise) about point J
, then translation along the vector, mapping point H
to point Z
rotation negative 90 degrees (clockwise) about point upper J , then translation along the vector, mapping point upper H to point upper Z

translation along the vector, mapping point J
to point P
, then rotation 90°
(counterclockwise) about point J
translation along the vector, mapping point upper J to point upper P , then rotation 90 degrees (counterclockwise) about point upper J

translation along the vector, mapping point G
to point N
, then rotation −90°
(clockwise) about point G
translation along the vector, mapping point upper G to point upper N , then rotation negative 90 degrees (clockwise) about point upper G

translation along the vector, mapping point J
to point P
, then rotation −90°
(clockwise) about point J

Answer 1

To determine which rigid motion would map triangle \( \triangle GHJ \) to triangle \( \triangle ZNP \), we consider the properties of rigid motions, specifically rotations and translations.

First, let's analyze the triangles:
- In \( \triangle GHJ \), assume \( G \) corresponds to point \( Z \), \( H \) corresponds to point \( N \), and \( J \) corresponds to point \( P \).

Since we need to find a sequence of motions that maps \( J \) to \( P \), we should:
1. Translate point \( J \) to point \( P \).
2. Rotate the triangle such that the orientation changes to match the orientation of \( \triangle ZNP \).

Let's examine the options:

1. **Rotation -90° (clockwise) about point J, then translation along the vector, mapping point H to point Z**:
- This seems plausible because the rotation will change the orientation effectively, but we first need to ensure \( J \) is correctly positioned at \( P \).

2. **Translation along the vector, mapping point J to point P, then rotation 90° (counterclockwise) about point J**:
- This would not result in the correct orientation as we are rotating counterclockwise instead.

3. **Translation along the vector, mapping point G to point N, then rotation -90° (clockwise) about point G**:
- This does not start with the translation of point \( J \) to point \( P \).

4. **Translation along the vector, mapping point J to point P, then rotation 90° (counterclockwise) about point J**:
- Same issue as option 2 with an incorrect rotation direction.

5. **Translation along the vector, mapping point G to point N, then rotation -90° (clockwise) about point G**:
- As previously stated, this point doesn't help in positioning \( J \) at \( P \).

6. **Translation along the vector, mapping J to point P, then rotation -90° (clockwise) about point J**:
- This will correctly position \( J \) at \( P \) and then rotate the triangle \( -90° \) clockwise about point \( P \) (which is now \( J \)), yielding the correct orientation.

The correct answer is:

**Translation along the vector, mapping point J to point P, then rotation -90° (clockwise) about point J.**