Question

7-22. on graph paper, graph $\triangle abc$ if its vertices are $a(-2, 7)$, $b(-5, 8)$, and $c(-3, 1)$.

a. reflect $\triangle abc$ across the $x$-axis to form $\triangle abc$. name the coordinates of each new vertex.

b. now rotate $\triangle abc$ $180^\circ$ about the origin $(0, 0)$ to form $\triangle abc$. name the coordinates of each new vertex.

c. describe a single transformation that would map $\triangle abc$ to $\triangle abc$.

use the etool below to solve the problem.
click the link at the right to view full version of the etool: 7-22 hw etool

Explanation:

Part (a) - Reflect over x - axis

Step 1: Recall reflection over x - axis rule

The rule for reflecting a point \((x,y)\) across the \(x\) - axis is \((x,y)\to(x, - y)\).

Step 2: Apply the rule to each vertex

• For \(A(-2,7)\): Using the rule \((x,y)\to(x, - y)\), we get \(A'(-2,-7)\).
• For \(B(-5,8)\): Using the rule \((x,y)\to(x, - y)\), we get \(B'(-5,-8)\).
• For \(C(-3,1)\): Using the rule \((x,y)\to(x, - y)\), we get \(C'(-3,-1)\).

Part (b) - Rotate \(180^{\circ}\) about the origin

Step 1: Recall \(180^{\circ}\) rotation rule

The rule for rotating a point \((x,y)\) \(180^{\circ}\) about the origin is \((x,y)\to(-x,-y)\).

Step 2: Apply the rule to \(A'\), \(B'\), \(C'\)

• For \(A'(-2,-7)\): Using the rule \((x,y)\to(-x,-y)\), we have \(-x = -(-2)=2\) and \(-y=-(-7) = 7\), so \(A''(2,7)\).
• For \(B'(-5,-8)\): Using the rule \((x,y)\to(-x,-y)\), we have \(-x=-(-5) = 5\) and \(-y=-(-8)=8\), so \(B''(5,8)\).
• For \(C'(-3,-1)\): Using the rule \((x,y)\to(-x,-y)\), we have \(-x=-(-3)=3\) and \(-y=-(-1) = 1\), so \(C''(3,1)\).

Part (c) - Single transformation

Step 1: Analyze the coordinates of \(A\) and \(A''\), \(B\) and \(B''\), \(C\) and \(C''\)

• \(A(-2,7)\) and \(A''(2,7)\): The \(x\) - coordinate changes sign (\(-2\to2\)) and the \(y\) - coordinate remains the same.
• \(B(-5,8)\) and \(B''(5,8)\): The \(x\) - coordinate changes sign (\(-5\to5\)) and the \(y\) - coordinate remains the same.
• \(C(-3,1)\) and \(C''(3,1)\): The \(x\) - coordinate changes sign (\(-3\to3\)) and the \(y\) - coordinate remains the same.

Step 2: Identify the transformation

The rule for reflecting a point \((x,y)\) across the \(y\) - axis is \((x,y)\to(-x,y)\). Comparing with the change from \(A\), \(B\), \(C\) to \(A''\), \(B''\), \(C''\) (where \(x\) - coordinate changes sign and \(y\) - coordinate stays the same), the single transformation is a reflection across the \(y\) - axis.

Answer:

s:

Part (a)

\(A'(-2,-7)\), \(B'(-5,-8)\), \(C'(-3,-1)\)

Part (b)

\(A''(2,7)\), \(B''(5,8)\), \(C''(3,1)\)

Part (c)

A reflection across the \(y\) - axis.