Chapter 4.5, Problem 32SPR

To determine

To evaluate: whether $\Delta ABC\cong \Delta XYZ$ .

Answer to Problem 32SPR

Yes,

  $\Delta ABC\cong \Delta XYZ$ .

Explanation of Solution

Given:

  $A\left(0,5\right),B\left(0,0\right),C\left(-2,0\right),X\left(4,8\right),Y\left(4,3\right),Z\left(6,3\right)$

Using distance formula to find AB, BC and AC

  $\begin{array}{c}AB=\sqrt{{\left(0-5\right)}^2+{\left(0-0\right)}^2}\\ =\sqrt{25+0}\\ =\sqrt{25}\\ =5\end{array}$

Similarly,

  $\begin{array}{c}BC=\sqrt{{\left(0-0\right)}^2+{\left(-2-0\right)}^2}\\ =\sqrt{0+4}\\ =\sqrt{4}\\ =2\end{array}$

Similarly,

  $\begin{array}{c}AC=\sqrt{{\left(5-0\right)}^2+{\left(0+2\right)}^2}\\ =\sqrt{25+4}\\ =\sqrt{29}\end{array}$

Similarly,

Using distance formula to find XY, YZ and ZX

  $\begin{array}{c}XY=\sqrt{{\left(3-8\right)}^2+{\left(4-4\right)}^2}\\ =\sqrt{25+0}\\ =\sqrt{25}\end{array}$

Similarly,

  $\begin{array}{c}YZ=\sqrt{{\left(3-3\right)}^2+{\left(6-4\right)}^2}\\ =\sqrt{0+4}\\ =\sqrt{4}\\ =2\end{array}$

Similarly,

  $\begin{array}{c}ZX=\sqrt{{\left(4-6\right)}^2+{\left(8-3\right)}^2}\\ =\sqrt{4+25}\\ =\sqrt{29}\end{array}$

It can be seen that,

  $AB=XY,BC=YZ\text{ and A}C=ZX$

Yes,

By SSS criterion,

  $\Delta ABC\cong \Delta XYZ$ .