# Common triangles

Last updated: 15 Nov 2008

Although we have, in theory, covered everything you need to know about triangles it is also worth familiarizing yourself with some of the triangles that turn up on a regular basis in the GMAT.

### 3-4-5 triangle

This is a right-angled triangle with sides of length 3, 4 and 5.

You can check that it is a right-angled using the Pythagoras theorem i.e. does $$3^{2} + 4^{2} = 5^{2}$$ ?

This triangle appears in the GMAT because the lengths of its sides are integer values and so it easy to work with it without the aid of a calculator.

This triangle will often turn up in disguise, for example all the triangles below are basically the 3-4-5 triangle, they have just been scaled up or down.

The 30-40-50 triangle is 10 times as big as the 3-4-5, the 12-16-20 triangle is 4 times as big and the 1.5-2-2.5 triangle is half the size.

### 5-12-13 triangle

This is a right-angled triangle with sides of length 5, 12 and 13.

You can check that it is a right-angled using Pythagoras i.e. does $$5^{2} + 12^{2} = 13^{2}$$ ?

This is much less common in the GMAT than the 3-4-5 triangle but is the only other distinct right-angled triangle which has small integer sides and for this reason it also appears in the GMAT.

### Isosceles right-angled triangle

This right-angled triangle with two sides the same length is a GMAT favorite because you can work out all the angles in it.

Since the two unknown angles are equal they must be 45°.

Also if we assume that the short sides have length 1, what is the length of the hypotenuse?

Set the hypotenuse to be $$x$$ and get practicing your Pythagoras Theorem.

$\begin{split} 1^2 + 1^2 &= x^2 // 2 &= x^2 // \sqrt{2} &= x \end{split}$

So length of the hypotenuse would $$\sqrt{2}$$, which gives us the ratios between the sides as 1:1:$$\sqrt{2}$$

### 30° - 60° - 90° triangle

This is the triangle formed when you cut an equilateral triangle in half (see below) and is a GMAT favorite because you know all the angles in it.

We can also calculate the ratios between the lengths of the sides for this triangle.

Assume the hypotenuse of the 30° - 60° - 90° triangle is 2.

The base is 1 (since it is half of one of the sides of the equilateral triangle).

We can use the Pythagoras theorem to find the length of the final side.

Set the height to be $$x$$ and we already know the base is 1 and the hypotenuse is 2.

$\begin{split} 1^2 + x^2 &= 2^2 \\ 1 + x^2 &= 4 \\ x^2 &= 3 \\ x &= \sqrt{3} \end{split}$

Which gives us sides of lengths 1, $$\sqrt{3}$$ and 2.