Chapter 8.2, Problem 35PPS

To determine

To Prove:

Theorem $8.5$ ( $△\mathrm{L}\mathrm{N}\mathrm{M}$ is right triangle)

Explanation of Solution

Give information:

In $△\mathrm{L}\mathrm{M}\mathrm{N}$ , $\mathrm{L}\mathrm{M}$ is longest side ${\mathrm{c}}^2={\mathrm{a}}^2+{\mathrm{b}}^2$

Proof:

Initially draw $△\mathrm{R}\mathrm{Q}\mathrm{P}$ with side lengths $\mathrm{a},\mathrm{b}$ and hypotenuse $\mathrm{x}$ .

Then ${\mathrm{x}}^2={\mathrm{a}}^2+{\mathrm{b}}^2$ by the Pythagoras theorem.

It is given that ${\mathrm{c}}^2={\mathrm{a}}^2+{\mathrm{b}}^2$ so by the solution property of equality, ${\mathrm{x}}^2={\mathrm{c}}^2$ .

By a property of square roots, $\mathrm{x}=\mathrm{c}$ .

  $△\mathrm{R}\mathrm{Q}\mathrm{P}\cong △\mathrm{L}\mathrm{M}\mathrm{N}$ by the $\mathrm{S}\mathrm{S}\mathrm{S}$ congruence postulate.

Corresponding parts of congruent triangles are congruent so $\mathrm{m}\angle \mathrm{R}=\mathrm{m}\angle \mathrm{N}=90$ .

By definition, $\angle \mathrm{N}$ is a right angle and $△\mathrm{L}\mathrm{M}\mathrm{N}$ is a right triangle.