Chapter 8.4, Problem 30WE

To determine

To prove: Hypotenuse $=\sqrt{2}.leg$ in a right angle triangle with angles ${90}^{\circ },{45}^{\circ }\text{and}{45}^{\circ }$ .

Explanation of Solution

Given information:

Right angle triangle with ${90}^{\circ },{45}^{\circ }\text{and}{45}^{\circ }$ .

  

Proof: From Pythagorean Theorem, ${\left(\text{hypotenuse}\right)}^{\text{2}}\text{=}{\left(\text{altitude}\right)}^{\text{2}}\text{+}{\left(\text{base}\right)}^{\text{2}}$

  $\begin{array}{l}\Rightarrow {\left(\text{hypotenuse}\right)}^2=a^2+a^2\\ \Rightarrow \text{hypotenuse}=\sqrt{2a^2}\\ \therefore \text{hypotenuse}=\sqrt{2}\times \text{leg}\end{array}$

Hence proved.