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

Recall that the hypotenuse is the side opposite the \(\displaystyle 90^\circ\) angle. 

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

\(\displaystyle H=2\cdot SS\)

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

\(\displaystyle 5\cdot2 = 10\)

A triangle with a \(\displaystyle 1:2:3\) angle relation is a \(\displaystyle 30\), \(\displaystyle 60\), \(\displaystyle 90\) degree triangle. The side opposite the smallest angle of a triangle is the shortest side, of length \(\displaystyle 3\). The side opposite the largest angle is the longest side, measuring twice the length of the shortest side for this triangle, \(\displaystyle 6\) units.

\(\displaystyle \frac{1}{2}=\sin{30}\)

\(\displaystyle =\frac{opp}{hyp}=\frac{3}{h}\)

Therefore, to make the above statement true \(\displaystyle h=6\).

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, 

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

so..

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

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

Therefore,

\(\displaystyle h=15\cdot2\)

\(\displaystyle h=30\)

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 \(\displaystyle 4\sqrt{3}\).

To double check the answer use the Pythagorean Thereom:

\(\displaystyle \\(4\sqrt{3})^2+4^2= \\16(3)+16= \\48+16=64 \\ \sqrt{64}=8\)

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

The shortest side is opposite the \(\displaystyle 30^\circ\) angle. Let's say that this side has length \(\displaystyle 1\).

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

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

\(\displaystyle 2\) is the hypotenuse, which is twice as big as the shortest side..

The ratio of the short side to the hypotenuse is \(\displaystyle 1:2\)

We know that in a 30-60=90 triangle, the smallest side corresponds to the side opposite the 30 degree angle.

Additionally, we know that the hypotenuse is 2 times the value of the smallest side, so in this case, that is 10.

The formula for 

\(\displaystyle \sin=\frac{opposite}{hypotenuse}\), so \(\displaystyle \frac{5}{10}\) or \(\displaystyle \frac{1}{2}\).

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

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

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

The height is just the smallest side, \(\displaystyle 6\).

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