Question

Find the value of $x$ . Write your answer in simplest form.

A right triangle is shown. The measure of one of the interior angles is 45 degrees. The length of one of the legs is 1 unit. The length of the hypotenuse is x units.

$x=$

Answer 1

The length of the hypotenuse is $x=\sqrt{2}$ units. ($x=\√2$)

Explanation:

If one of the angles of a right triangle is 45 degrees, then the other acute angle is also 45 degrees. Let's label the other leg of the triangle as $y$ units.

Using the Pythagorean Theorem, we have:

$1^2 + y^2 = x^2$

Simplifying:

$1 + y^2 = x^2$

We can also use the fact that the ratio of the sides of a 45-45-90 triangle is $1:1:\sqrt{2}$ to write:

$\frac{x}{1}=\sqrt{2}$

Simplifying:

$x=\sqrt{2}$

Now, substituting this value of $x$ into our equation $1 + y^2 = x^2$, we have:

$1 + y^2 = (\sqrt{2})^2$

Simplifying:

$1 + y^2 = 2$

$y^2 = 1$

$y = 1$ (since we're looking for a positive length)

Therefore, the length of the hypotenuse is $x=\sqrt{2}$ units.

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