Properties of 3-4-5 Triangles: Definition and Uses

Are you up to speed on your Pythagorean triples? If you could use a little help with that, watch this lesson to learn what 3-4-5 triangles are, how they’re used, and why they’re important.
Pythagorean Theorem
Once upon a time, a famous Greek mathematician called Pythagoras proved a formula for figuring out the third side of any right triangle if you know the other two sides.

A right triangle is any triangle with one right (90-degree) angle. Usually this is indicated by putting a little square marker inside the right triangle. In any right triangle, the two sides bordering on the right angle will be shorter than the side opposite the right angle, which will be the longest side, or hypotenuse. If we call the short sides a and b and the long side c, then the Pythagorean Theorem states that:

a^2 + b^2 = c^2

It doesn’t matter which of the two shorter sides is a and which is b. It only matters that the longest side always has to be c.

Let’s take a look at how this works in practice. Say we have a triangle where the two short sides are 4 and 6. We don’t know what the long side is but we can see that it’s a right triangle. To find the long side, we can just plug the side lengths into the Pythagorean theorem. 4 squared plus 6 squared equals c squared. That means c squared equals 60, and c is equal to the square root of 60, or approximately 7.746.

Pythagorean Triples
But what does this all have to do with 3, 4, and 5? Well, you might notice that 7.746 isn’t a very nice number to work with. You probably wouldn’t want to do a lot of calculations with that, and your teachers probably don’t want to, either! Wouldn’t it be nicer to have a triangle with easy side lengths, like, say, 3, 4, and 5?

That’s where the Pythagorean triples come in. A Pythagorean triple is a right triangle where all the sides are integers. And – you guessed it – one of the most popular Pythagorean triples is the 3-4-5 right triangle. 3 and 4 are the lengths of the shorter sides, and 5 is the length of the hypotenuse, the longest side opposite the right angle.

If you run through the Pythagorean Theorem on this one, you can see that it checks out:

3^2 + 4^2 = 5^2

9 + 16 = 25

25 = 25

3-4-5 Multiples
It’s not just 3, 4, and 5, though. You can scale this same triplet up or down by multiplying or dividing the length of each side. For example, a 6-8-10 triangle is just a 3-4-5 triangle with all the sides multiplied by 2. As long as you multiply each side by the same number, all the side lengths will still be integers and the Pythagorean Theorem will still work. On the other hand, you can’t add or subtract the same number to all sides. There’s no such thing as a 4-5-6 triangle.

If you applied the Pythagorean Theorem to this, you’d get –

4^2 + 5^2 = 6^2

16 + 25 = 36

41 = 36

– which isn’t true. You can absolutely have a right triangle with short sides 4 and 5, but the hypotenuse would have to be the square root of 41, which is approximately 6.4.

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