Question

Find the exact length, in units, of the hypotenuse of the right triangle, shown below. Write your answer in simplified radical form.

Answer 1

Right Triangles

A right triangle is identified because it has an angle of 90° (marked as a little square).

In a right triangle, there is a larger side called the hypotenuse, and two shorter sides called the legs.

Each one of the acute angles in a right triangle has an adjacent leg and an opposite leg. For example, the angle of 45° given in the figure has 15 as the adjacent leg. The hypotenuse is below it, and we'll call it H.

There is a trigonometric ratio called the cosine that relates the adjacent leg of an angle with the hypotenuse as follows:

`\displaystyle\cos \theta=\frac{\text{adjacent leg}}{\text{hypotenuse}}`

Applying to the given triangle:

`\cos 45^o=(15)/(H)`

Solving for H:

`H=(15)/(\cos45^o)=\frac{15}{\frac{\sqrt[]{2}}{2}}=15\cdot\frac{2}{\sqrt[]{2}}\cdot\frac{\sqrt[]{2}}{\sqrt[]{2}}=15\sqrt[]{2}`

Hypotenuse length:

`15\sqrt[]{2}`