Chapter 10.4, Problem 48HP

To determine

To explain:

The method to estimate the lengths of right triangle’s legs

Answer to Problem 48HP

The length of each leg $45°-45°-90°$ of the triangle can be found by multiplying the radius of the circle in which it is inscribed by $\sqrt{2}$

Explanation of Solution

Given information:

A $45°-45°-90°$ right triangle is inscribed in a circle

Here, given that:

In $45°-45°-90°$ triangle, one angle is $90°$ and other two are inscribed as $45°$ . Here, the hypotenuse is across from the right angle.

Know that:

When the angle of the triangle is right angle then the angle of a triangle intercepts a diameter.

And

  $\sin \theta =\frac{\text{Opposite}}{\text{Hypotenuse}}$

Therefore, one can say that the hypotenuse of right angle triangle is a diameter and has a length of 2r.

To estimate the lengths of right triangle’s legs need to use the trigonometry:

  $\begin{array}{l}\sin 45°=\frac{Leg}{2r}\\ \frac{1}{\sqrt{2}}=\frac{Leg}{2r}\\ Leg=\frac{2r}{\sqrt{2}}\\ Leg=\sqrt{2}r\end{array}$