Special triangles are right triangles that have special proportions for their sides. The 30°-60°-90° triangle has the proportions 1:√3:2. The 45°-45°-90° triangle has the proportions 1:1:√2. All the lengths of these sides can be easily found if we only know the length of one of the sides.

Here, we will learn more details about these special triangles. We will learn to derive its proportions and solve some practice exercises.

## Special triangle 45°-45°-90°

A 45°-45°-90° triangle is a right triangle that has two acute angles with a measure of 45°. This means that these triangles are also isosceles and have sides with special proportions.

The proportions of these triangles can be obtained using the Pythagorean theorem. Since these triangles have two angles with the same measurements, their opposite sides must be the same length. Therefore, using the Pythagorean theorem, we can assume that the two equal sides have a length of 1 unit, so the hypotenuse is:

$latex {{c}^2}={{a}^2}+{{b}^2}$

$latex {{c}^2}={{1}^2}+{{1}^2}$

$latex {{c}^2}=2$

$latex c=\sqrt{2}$

We can scale the triangle by a factor of *k*. Since we are only scaling the triangle without changing any of its angles, we get the proportions of the lengths of the sides of the triangle 45°-45°-90°.

## Special triangle 30°-60°-90°

A 30°-60°-90° triangle is a right triangle that contains the acute angles 30° and 60°. The sides of these triangles have lengths with special proportions.

We can obtain these proportions using the Pythagorean theorem. For this, we can consider that the 30°-60°-90° triangle is half of an equilateral triangle as shown in the following diagram.

We can assume that one of the short sides of the 30°-60°-90° triangle has a length of 1 unit. Therefore, the base of the equilateral triangle will measure 2 units and the hypotenuse of the 30°-60°-90° triangle will also measure 2 units since an equilateral triangle has all its sides with the same length.

With this, we can calculate the longitude of the second leg of the triangle 30 °-60°-90°:

$latex {{a}^2}+{{b}^2}={{c}^2}$

$latex {{1}^2}+{{b}^2}={{2}^2}$

$latex 1+{{b}^2}=4$

$latex {{b}^2}=3$

$latex b=\sqrt{3}$

Now, we can scale the triangle by a factor of *k*. When scaling, we are only changing the lengths of the sides, but we are keeping the angles and proportions. Therefore, we get the general proportions of the triangle 30°-60°-90°.

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## Solved problems of special triangles

### EXAMPLE 1

Determine the measure of side *x* in the triangle below.

**Solution:** This is a 30°-60°-90° triangle and we know that these triangles have the proportions 1:2:√3. The side that measures 5 units corresponds to the 30° angle and the side we are looking for corresponds to the 90° angle. Therefore, its proportion is 2 and we have:

$latex x = 5 \times 2$

$latex x = 10$

Side *x* is 10 units long.

### EXAMPLE 2

What is the measure of side *x* in the triangle below?

**Solution:** This is a 45°-45°-90° triangle and we know that these triangles have the proportions 1:1:√2. The side with the measure of 8 units corresponds to one of the legs opposite the 45° angle and the side we are looking for corresponds to the side opposite the 90° angle.

Therefore, the proportion of side *x* is √2. Then, we have:

$latex x = 8 \sqrt{2}$

### EXAMPLE 3

Determine the measure of *x* in the triangle below.

**Solution:** This is a 30°-60°-90° triangle. Therefore, we know that its proportions are 1:√3:2. We know the length of the side opposite the 90° angle, that is, the side that has proportion 2.

We want to find the measure of the side *x* that is opposite the 30° angle, that is, the side that has proportion 1. Therefore, we divide by 2 to find its measure:

$latex x = \frac{6 \sqrt{3}}{2}$

$latex x = 3 \sqrt{3}$

## See also

Interested in learning more about similar figures, congruency and other geometry topics? Take a look at these pages:

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