Chapter 4.7, Problem 17PPS

To determine

To graph:

Each pair of triangles with the following vertices and tell that the type of transformation by verification of the congruence transformation:

  $\begin{array}{l}\mathrm{M}\left(-7,-1\right),\mathrm{P}\left(-7,-7\right),\mathrm{R}\left(-1,-4\right);\\ \mathrm{T}\left(7,-1\right),\mathrm{V}\left(-7,-7\right),\mathrm{S}\left(1,-4\right)\end{array}$

Explanation of Solution

Given information:

  $\begin{array}{l}\mathrm{M}\left(-7,-1\right),\mathrm{P}\left(-7,-7\right),\mathrm{R}\left(-1,-4\right);\\ \mathrm{T}\left(7,-1\right),\mathrm{V}\left(-7,-7\right),\mathrm{S}\left(1,-4\right)\end{array}$

Graph:

  

Interpretation:

From the drawn graph of the triangle with the given vertices, understands that each vertex of triangle and its image are the same distance from the $\mathrm{y}$ -axis.

Therefore, it is reflection which implies that $\mathrm{\Delta }\mathrm{T}\mathrm{V}\mathrm{S}$ is reflection of $\mathrm{\Delta }\mathrm{M}\mathrm{P}\mathrm{R}$ . Use the distance formula to find the length of each side of the triangles $\mathrm{\Delta }\mathrm{T}\mathrm{V}\mathrm{S}$ and $\mathrm{\Delta }\mathrm{M}\mathrm{P}\mathrm{R}$

The distance formula is given by the following expression:

  $\mathrm{d}=\sqrt{{\left({\mathrm{x}}_2-{\mathrm{x}}_1\right)}^2+{\left({\mathrm{y}}_2-{\mathrm{y}}_1\right)}^2}$

In the triangle $\mathrm{\Delta }\mathrm{T}\mathrm{V}\mathrm{S}$ , the side lengths are calculated as:

  $\begin{array}{l}\overline{\mathrm{T}\mathrm{V}}=\sqrt{{\left(7-7\right)}^2+{\left(-7-\left(-1\right)\right)}^2}=\sqrt{36}=6\\ \overline{\mathrm{V}\mathrm{S}}=\sqrt{{\left(1-7\right)}^2+{\left(-4-\left(-7\right)\right)}^2}=\sqrt{45}\\ \overline{\mathrm{S}\mathrm{T}}=\sqrt{{\left(7-1\right)}^2+{\left(-1-\left(-4\right)\right)}^2}=\sqrt{45}\end{array}$

In the triangle $\mathrm{\Delta }\mathrm{M}\mathrm{P}\mathrm{R}$ , the side lengths are calculated as:

  $\begin{array}{l}\overline{\mathrm{M}\mathrm{P}}=\sqrt{{\left(-7-\left(-7\right)\right)}^2+{\left(-7-\left(-1\right)\right)}^2}=\sqrt{36}=6\\ \overline{\mathrm{P}\mathrm{R}}=\sqrt{{\left(-1-\left(-7\right)\right)}^2+{\left(-4-\left(-7\right)\right)}^2}=\sqrt{45}\\ \overline{\mathrm{R}\mathrm{M}}=\sqrt{{\left(-7-\left(-1\right)\right)}^2+{\left(-1-\left(-4\right)\right)}^2}=\sqrt{45}\end{array}$

From the above calculations:

  $\begin{array}{l}\overline{\mathrm{T}\mathrm{V}}\cong \overline{\mathrm{M}\mathrm{P}}\\ \overline{\mathrm{V}\mathrm{S}}\cong \overline{\mathrm{P}\mathrm{R}}\\ \overline{\mathrm{S}\mathrm{T}}\cong \overline{\mathrm{R}\mathrm{M}}\end{array}$

Therefore, each pair of corresponding sides has the same measure that means they are congruent by SSS theorem.

Therefore,

  $\mathrm{\Delta }\mathrm{T}\mathrm{V}\mathrm{S}\cong \mathrm{\Delta }\mathrm{M}\mathrm{P}\mathrm{R}$