Chapter 8, Problem 16CR

To determine

To calculate: the length of the hypotenuse and altitude to the hypotenuse of an isosceles triangle.

Answer to Problem 16CR

The length of hypotenuse of the isosceles triangle is $12\sqrt{2}$

The altitude to the hypotenuse of the triangle is $6\sqrt{2}$

Explanation of Solution

Given Information: The legs of an isosceles right triangle are 12 units long.

Formula used:

1) The relationship between the legs of right triangle (shown in figure below) a , b and its hypotenuse c is obtained by using Pythagoras theorem.

  

It can be expressed as,

  $\begin{array}{l}{c}^2=a^2+b^2\\ c=\sqrt{a^2+b^2}\end{array}$

2) Also, in a right triangle the sine of the angle is the ratio of the measure of side opposite to the angle to the hypotenuse.

It can be expressed as,

  $\sin B=\frac{b}{c}$

Calculation: Consider the isosceles triangle given below.

  

Here, two right triangles are formed with in the isosceles triangle as shown in the above Figure.

In the triangle ABC, the Pythagoras theorem gives,

  $\begin{array}{c}BC=\sqrt{{12}^2+{12}^2}\\ =\sqrt{144+144}\\ BC=12\sqrt{2}\end{array}$

Now, in ΔABD,

  $\begin{array}{c}\sin 45°=\frac{x}{12}\\ \frac{1}{\sqrt{2}}=\frac{x}{12}\\ x=\frac{12}{\sqrt{2}}\\ =6\sqrt{2}\end{array}$