Question Video: The Side Lengths of 45-45-90 Triangles

Find, in terms of 𝑥, the length of the hypotenuse of this triangle.

04:29

Video Transcript

Find, in terms of 𝑥, the length of the hypotenuse of this triangle.

From the diagram, we can see that we have a right-angled triangle. The hypotenuse of a right-angled triangle is its longest side, the side opposite the right angle. We’ll give this side the letter ℎ for use in our working out. The other thing we can see about this right-angled triangle is that it’s also an isosceles triangle as the length of the two shorter sides are the same; they’re both 𝑥 units.

We were asked to find the length of the hypotenuse. And there are two approaches that we could take. So we’ll use both. The first approach is that as this is a right-angled triangle, we’ll apply the Pythagorean theorem.

The Pythagorean theorem tells us that in a right angle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. In this triangle, this means that ℎ squared is equal to 𝑥 squared plus 𝑥 squared. This can be simplified to give ℎ squared is equal to two 𝑥 squared.

To find an expression for ℎ, we now need to take the square root of both sides. We have that ℎ is equal to the square root of two 𝑥 squared. Now, the laws of surds tell us that we can separate the square root of a product into the product of the individual square roots. So we can express ℎ as the square root of two multiplied by the square root of 𝑥 squared. The square root of 𝑥 squared is just 𝑥.

So we have our simplified expression for the length of the hypotenuse in terms of 𝑥: ℎ is equal to 𝑥 root two. So this is our first approach to this question by applying the Pythagorean theorem.

Our second approach is going to be to apply some trigonometry. As this triangle is isosceles, then the size of the two non-right angles are each 45 degrees. And 45 degrees is one of the special angles for which the trigonometric ratios can be expressed in terms of surds.

Sine and cosine of 45 degrees are both the same. They’re both equal to root two over two. Tan of 45 degrees is equal to one. Remember these values are the ratios between different pairs of sides in the triangle. And so we can use these values in order to find the value of ℎ.

Let’s consider just one of the 45-degree angles. And I’ve labelled the three sides of the triangle in relation to this. We have the opposite, the adjacent, and the hypotenuse. Let’s use the ratio between the opposite and the hypotenuse. SOHCAHTOA tells us that this is the sine ratio in this triangle.

By recalling the definition of the sine ratio as the opposite divided by hypotenuse, we see that sine of 45 degrees is equal to 𝑥 over ℎ. Remember sine of 45 degrees is equal to root two over two. So we can now substitute this value into the ratio. We now have that root two over two is equal to 𝑥 over ℎ. And we’d like to rearrange this equation in order to give ℎ in terms of 𝑥.

The first step is to cross multiply. This has the effect of eliminating the denominators in the two fractions and gives ℎ root two is equal to two 𝑥. Next, we need to divide both sides of the equation by root two. This gives ℎ is equal to two 𝑥 over root two.

Now, this looks like a different expression for the hypotenuse from the one we found earlier. But this is because there’s a surd in the denominator, which we need to rationalize. To do so, we multiply by root two over root two, a fraction which is equivalent to one. This gives two 𝑥 root two in the numerator and just two in the denominator. The factors of two in the numerator and denominator will cancel each other out, leaving 𝑥 root two, the same answer as before.

So we’ve used two different approaches: one applying the Pythagorean theorem and the other applying the exact trigonometric value of sin 45 degrees to show that the length of the hypotenuse is 𝑥 root two.

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