Chapter 4.6, Problem 57SPR

To determine

Whether triangle STU is congruent to triangle XYZ.

Answer to Problem 57SPR

The triangles are not congruent.

Explanation of Solution

Given:

The given vertices are:

  $S\left(0,5\right),T\left(0,0\right),U\left(1,1\right),X\left(4,8\right),Y\left(4,3\right),Z\left(6,3\right)$

Calculation:

Finding the length of each side of triangle STU:

The length of ST will be:

  $\begin{array}{l}ST=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y{1}_{}\right)}^2}\\ ST=\sqrt{{\left(0-0\right)}^2+{\left(0-5\right)}^2}\\ ST=\sqrt{0+25}\\ ST=5\end{array}$

The length of TU will be:

  $\begin{array}{l}TU=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y{1}_{}\right)}^2}\\ TU=\sqrt{{\left(1-0\right)}^2+{\left(1-0\right)}^2}\\ TU=\sqrt{1+1}\\ TU=\sqrt{2}\end{array}$

The length of SU will be:

  $\begin{array}{l}SU=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y{1}_{}\right)}^2}\\ SU=\sqrt{{\left(1-0\right)}^2+{\left(1-5\right)}^2}\\ SU=\sqrt{1+16}\\ SU=\sqrt{17}\end{array}$

Finding the length of each side of triangle XYZ:

The length of XY will be:

  $\begin{array}{l}XY=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y{1}_{}\right)}^2}\\ XY=\sqrt{{\left(4-4\right)}^2+{\left(3-8\right)}^2}\\ XY=\sqrt{0+25}\\ XY=5\end{array}$

The length of YZ will be:

  $\begin{array}{l}YZ=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y{1}_{}\right)}^2}\\ YZ=\sqrt{{\left(6-4\right)}^2+{\left(3-3\right)}^2}\\ YZ=\sqrt{4+0}\\ YZ=2\end{array}$

The length of ZX will be:

  $\begin{array}{l}ZX=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y{1}_{}\right)}^2}\\ ZX=\sqrt{{\left(6-4\right)}^2+{\left(3-8\right)}^2}\\ ZX=\sqrt{4+25}\\ ZX=\sqrt{29}\end{array}$

As the corresponding sides of the two triangles are not congruent, hence the triangles are not congruent.