Chapter 4.7, Problem 19PPS

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:

  $\begin{array}{l}\mathrm{A}\left(-4,5\right),\mathrm{B}\left(0,2\right),\mathrm{C}\left(-4,2\right);\\ \mathrm{X}\left(-5,-4\right),\mathrm{Y}\left(-2,0\right),\mathrm{Z}\left(-2,-4\right)\end{array}$

Explanation of Solution

Given information:

  $\begin{array}{l}\mathrm{A}\left(-4,5\right),\mathrm{B}\left(0,2\right),\mathrm{C}\left(-4,2\right);\\ \mathrm{X}\left(-5,-4\right),\mathrm{Y}\left(-2,0\right),\mathrm{Z}\left(-2,-4\right)\end{array}$

Graph:

  

Interpretation:

From the drawn graph of the triangle with the given vertices, understands that $\mathrm{\Delta }\mathrm{X}\mathrm{Y}\mathrm{Z}$ is rotation of $\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}$ . Use the distance formula to find the length of each side of the triangles $\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}$ and $\mathrm{\Delta }\mathrm{X}\mathrm{Y}\mathrm{Z}$

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 $\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}$ , the side lengths are calculated as:

  $\begin{array}{l}\overline{\mathrm{A}\mathrm{B}}=\sqrt{{\left(0+4\right)}^2+{\left(2-5\right)}^2}=\sqrt{16+9}=5\\ \overline{\mathrm{B}\mathrm{C}}=\sqrt{{\left(-4-0\right)}^2+{\left(2-2\right)}^2}=\sqrt{16}=4\\ \overline{\mathrm{A}\mathrm{C}}=\sqrt{{\left(-4+4\right)}^2+{\left(2-5\right)}^2}=\sqrt{9}=3\end{array}$

In the triangle $\mathrm{\Delta }\mathrm{X}\mathrm{Y}\mathrm{Z}$ , the side lengths are calculated as:

  $\begin{array}{l}\overline{\mathrm{X}\mathrm{Y}}=\sqrt{{\left(-2+5\right)}^2+{\left(0+4\right)}^2}=\sqrt{9+16}=5\\ \overline{\mathrm{Y}\mathrm{Z}}=\sqrt{{\left(-2+2\right)}^2+{\left(-4-0\right)}^2}=\sqrt{16}=4\\ \overline{\mathrm{X}\mathrm{Z}}=\sqrt{{\left(-2+5\right)}^2+{\left(-4+4\right)}^2}=\sqrt{9}=3\end{array}$

From the above calculations:

  $\begin{array}{l}\overline{\mathrm{A}\mathrm{B}}\cong \overline{\mathrm{X}\mathrm{Y}}\\ \overline{\mathrm{B}\mathrm{C}}\cong \overline{\mathrm{Y}\mathrm{Z}}\\ \overline{\mathrm{A}\mathrm{C}}\cong \overline{\mathrm{X}\mathrm{Z}}\end{array}$

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

Therefore,

  $\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}\cong \mathrm{\Delta }\mathrm{X}\mathrm{Y}\mathrm{Z}$