Chapter 4, Problem 38SGR

To determine

To graph:

Original figure and identification of transformation and verify its congruence transformation.

Explanation of Solution

Given information:

  $A\left(1,1\right),B\left(2,3\right),C\left(3,-1\right),M\left(-1,1\right),N\left(-2,3\right),O\left(-3,-1\right)$

Graph:

  

Interpretation:

A reflection is a transformation over a line. Each point of the preimage and its image are the same distance from the line of reflection.

In the given figure, each vertex and its image are same distance from the $y-$ axis. hence, $△MNO$ is a reflection of $△ABC$ .

If three sides of one triangle are congruent to three sides of second triangle, then the triangles are congruent.

Initially find he length of each side of $△ABC$ .

Find the length of $\overline{AB}$ with $A\left(1,1\right),B\left(2,3\right)$

  $\begin{array}{l}AB=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ AB=\sqrt{{\left(2-1\right)}^2+{\left(3-1\right)}^2}\\ AB=\sqrt{1+4}\\ AB=\sqrt{5}\end{array}$

Find the length of $\overline{BC}$ with $B\left(2,3\right),C\left(3,-1\right)$

  $\begin{array}{l}BC=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ BC=\sqrt{{\left(3-2\right)}^2+{\left(-1-3\right)}^2}\\ BC=\sqrt{1+16}\\ BC=\sqrt{17}\end{array}$

Find the length of $\overline{CA}$ with $C\left(3,-1\right),A\left(1,1\right)$

  $\begin{array}{l}CA=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ CA=\sqrt{{\left(1-3\right)}^2+{\left(1-\left(-1\right)\right)}^2}\\ CA=\sqrt{4+4}\\ CA=\sqrt{8}\end{array}$

Second find the length of each side $△MNO$ ,

Find the length of $\overline{MN}$ with $M\left(-1,1\right),N\left(-2,3\right)$

Find the length of $\overline{NO}$ with $N\left(-2,3\right),O\left(-3,-1\right)$

  $\begin{array}{l}NO=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ NO=\sqrt{{\left(-3-\left(-2\right)\right)}^2+{\left(-1-3\right)}^2}\\ CA=\sqrt{1+16}\\ CA=\sqrt{17}\end{array}$

Find the length of $\overline{OM}$ with $O\left(-3,-1\right),M\left(-1,1\right)$

  $\begin{array}{l}OM=\sqrt{{\left(x_2-x_1\right)}^2+\left(y_2-y_1\right)}\\ OM=\sqrt{{\left(-1-\left(-3\right)\right)}^2+{\left(1-\left(-1\right)\right)}^2}\\ OM=\sqrt{4+4}\\ OM=\sqrt{8}\end{array}$

Thus, $\overline{AB}\cong \overline{MN},\overline{BC}\cong \overline{NO},\overline{AC}\cong \overline{MO}$