Questions: Triangle ABC is reflected about the line y=-x to give triangle A'B'C' with vertices A'(-1,1), B'(-2,-1), C(-1,0). What are the vertices of triangle ABC? A. A(1,-1), B(-1,-2), C(0,-1) B. A(-1,1), B(1,2), C(0,1) C. A(-1,-1), B(-2,-1), C(-1,0) D. A(1,1), B(2,-1), C(1,0) E. A(1,2), B(-1,1), C(0,1)

Triangle ABC is reflected about the line y=-x to give triangle A'B'C' with vertices A'(-1,1), B'(-2,-1), C(-1,0). What are the vertices of triangle ABC?
A. A(1,-1), B(-1,-2), C(0,-1)
B. A(-1,1), B(1,2), C(0,1)
C. A(-1,-1), B(-2,-1), C(-1,0)
D. A(1,1), B(2,-1), C(1,0)
E. A(1,2), B(-1,1), C(0,1)

Transcript text: $\triangle A B C$ is reflected about the line $y=-x$ to give $\triangle A^{\prime} B^{\prime} C^{\prime}$ with vertices $A^{\prime}(-1,1), B^{\prime}(-2,-1), C(-1,0)$. What are the vertices of $\triangle A B C$ ? A. $A(1,-1), B(-1,-2), C(0,-1)$ B. $A(-1,1), B(1,2), C(0,1)$ C. $A(-1,-1), B(-2,-1), C(-1,0)$ D. $A(1,1), B(2,-1), C(1,0)$ E. $A(1,2), B(-1,1), C(0,1)$

Solution

Solution Steps

Step 1: Understand the reflection over the line $ y = -x $

Reflecting a point over the line $ y = -x $ swaps its coordinates and changes their signs. Mathematically, if a point $ P(x, y) $ is reflected over $ y = -x $, its image $ P' $ will have coordinates $ (-y, -x) $.

Step 2: Apply the reflection formula to find the original vertices

Given the reflected vertices $ A'(-1, 1) $, $ B'(-2, -1) $, and $ C'(-1, 0) $, we can find the original vertices $ A $, $ B $, and $ C $ by reversing the reflection process. For each reflected point $ (x', y') $, the original point $ (x, y) $ is given by: \[ (x, y) = (-y', -x') \]

For $ A'(-1, 1) $: \[ A = (-1, -(-1)) = (-1, 1) \]
For $ B'(-2, -1) $: \[ B = (-(-1), -(-2)) = (1, 2) \]
For $ C'(-1, 0) $: \[ C = (-0, -(-1)) = (0, 1) \]

Step 3: Compare the calculated vertices with the given options

The calculated vertices are: