Question

△ABC has vertices A(−7,−13), B(12,−8), and C(−17,19). Which of the following represents the reflection of △ABC across the line y=x and its rotation of 90∘ about the origin?

Answer 1

A (−7, −13) → A ′(−13, −7) → A ″(7, −13);

B (12, −8) → B ′(−8, 12) → B ″(−12, −8);

C (−17, 19) → C ′(19, −17) → C ″(17, 19)

Step-by-step explanation

The coordinates of the vertices of the preimage are given.

To find the image as it reflected from the preimage across the y=x line, use the transformation rule: (x,y)→(y,x).

Apply the transformation rule to vertices A(−7,−13), B(12,−8), and C(−17,19).

A(−7,−13)→A'(−13,−7).

B(12,−8)→B'(−8,12).

C(−17,19)→C'(19,−17).

To determine the vertices of the image after the rotation of 90∘ about the origin, use the rule: (x,y)→(−y,x).

Apply the rotation rule to the vertices of △A'B'C'.

A'(−13,−7)→A''(7,−13).

B'(−8,12)→B''(−12,−8).

C'(19,−17)→C''(17,19).

Therefore,

A(−7,−13)→A'(−13,−7)→A''(7,−13)

B(12,−8)→B'(−8,12)→B''(−12,−8)

C(−17,19)→C'(19,−17)→C''(17,19)

represents the reflection of △ABC across the line y=x and its rotation of 90∘ about the origin.

Answer 2

Option B

Explanation:

Plot points A, B, C and line y=x on the coordinate plane (see attached diagram, blue points)

1. The reflection across the line y=x has the rule

(x,y)→(y,x)

So,

• A(-7,-13)→A'(-13,-7)
• B(12,-8)→B'(-8,12)
• C(-17,19)→C'(19,-17)

Points A', B', C' are marked in red on the diagram

2. The rotation by 90° clockwise about the origin has the rule

(x,y)→(-y,x)

So,

• A'(-13,-7)→A''(7,-13)
• B'(-8,12)→B''(-12,-8)
• C'(19,-17)→C''(17,19)