Trigonometry - 30-60-90 Triangles

In a 30-60-90 triangle, the length of the side opposite the $30^\circ$ angle is . What is the length of the hypotenuse?

$10\sqrt{3}$

$5\sqrt{3}$

Explanation

By definition, the length of the hypotenuse is twice the length of the side opposite the $30^\circ$ angle.

Recall that the hypotenuse is the side opposite the $90^\circ$ angle.

Thus, using the equation below, where ss represents the short side (that opposite the $30^\circ$ angle) we get:

$H=2\cdot SS$

Plugging in our values for the short side we find the hypotenuse as follows:

$5\cdot2 = 10$

In a triangle, the side opposite the degree angle is . How long is the side opposite the degree angle?

$15\sqrt{2}$

$15\sqrt{3}$

Explanation

Based on the 30-60-90 identity, the measure of the side opposite the 30 degree angle is doubled to get the hypotenuse.

Therefore,

$h=15\cdot2$

It is known that for a 30-60-90 triangle,

$\sin(30\degree)=\frac{6}{12}$ .

Find the area of the triangle.

Note:

$\textup{Area}=\frac{1}{2}*b*h$

$18\sqrt3$

Explanation

First, we know that in a 30-60-90 triangle,

$\sin(30)=\frac{height}{hypotenuse}$ .

Also, the base is the smallest side times $\sqrt3$ , so in our case it is $6\sqrt3$ .

The height is just the smallest side, .

Substituting these values into the formula given for area of a triangle, we obtain the answer $18\sqrt3$ .

What is the ratio of the side opposite the $30^{\circ}$ angle to the hypotenuse?

$1: \sqrt {3}$

$2:\sqrt {3}$

Explanation

Step 1: Locate the side that is opposite the $30^{\circ}$ side..

The shortest side is opposite the $30^\circ$ angle. Let's say that this side has length .

Step 2: Recall the ratio of the sides of a $30^\circ-60^\circ-90^\circ$ triangle:

From the shortest side, the ratio is $1:\sqrt {3}:2$ .

is the hypotenuse, which is twice as big as the shortest side..

The ratio of the short side to the hypotenuse is

Triangle is equilateral with a side length of .

What is the height of the triangle?

$\frac{\sqrt3}{2}s$

$\frac{2s}{\sqrt3}$

$\frac{s}{2}$

$\sqrt{s}$

Explanation

An equilateral triangle has internal angles of 60°, so the sin of one of those angles is equivalent to the height of the triangle divided by the side length,

$\sin(60)=\frac{height}{side}$

so..

$h=s*\sin(60)=s\frac{\sqrt3}{2}$

A triangle has three angles , and such that and . The side opposite to measures units in length. How long is the side opposite of ?

$\sqrt3$

$3\sqrt3$

Explanation

A triangle with a angle relation is a , , degree triangle. The side opposite the smallest angle of a triangle is the shortest side, of length . The side opposite the largest angle is the longest side, measuring twice the length of the shortest side for this triangle, units.

$\frac{1}{2}=\sin{30}$

$=\frac{opp}{hyp}=\frac{3}{h}$

Therefore, to make the above statement true .

What is the height of an equilateral triangle with side length 8?

$4\sqrt{3}$

$8\sqrt{3}$

$4\sqrt{2}$

$\frac{4\sqrt{3}}{3}$

Explanation

The altitude of an equilateral triangle splits it into two 30-60-90 triangles. The height of the triangle is the longer leg of the 30-60-90 triangle. If the hypotenuse is 8, the longer leg is $4\sqrt{3}$ .