A 45°-45°-90° triangle has legs of length 5. What is the length of…

Step by step

The question

Question: find the hypotenuse of a 45-45-90 triangle with a leg of 5.

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Half of a square

The angles are 45°, 45° and 90°.

This is half of a square.

The longest side cuts the square in two.

So the two legs are sides of the square.

They are equal: both are 5.

2 Find the pattern

First a pattern for any size: call each leg x.

Pythagoras: c² = x² + x².

One x² and one more x² make 2x².

c times c is 2x².

Which number times itself is 2x²?

x√2.

x times x is x².

And √2 times √2 is 2: that is what √2 means.

So the hypotenuse is always a leg times √2.

3 Use the pattern

Here the leg is 5.

So c = 5 × √2.

We leave out the times sign.

So c = 5√2.

With a calculator, that is about 7.1.

Careful!

The hypotenuse is not the two legs added.

The way along the diagonal is shorter than going round two sides.

4 Check

Let's check with Pythagoras: a² + b².

Now c²: (5√2)² is 5√2 times 5√2.

That is 5 × 5 times √2 × √2: 25 times 2.

Both are 50.

So c = 5√2 is right!

5 Pick the choice

Our answer is c = 5√2.

That is choice B.

6 Why not the others

A is 5.

Wrong: that is a leg, not the hypotenuse.

C is 5√3.

Wrong: √3 is for a 30-60-90 triangle.

D is 10.

Wrong: doubled the leg: that is the 30-60-90 rule.

That is the Careful mistake from before.

Let's look back.

First: half of a square.

Then: find the pattern.

Then: use the pattern.

Then: check.

Then: pick the choice.

Last: why not the others.