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4.43: 30-60-90 Right Triangles

4.43: 30-60-90 Right Triangles

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Hypotenuse equals twice the smallest leg, while the larger leg is \(\sqrt{3}\) times the smallest.

One of the two special right triangles is called a 30-60-90 triangle, after its three angles.

30-60-90 Theorem: If a triangle has angle measures \(30^{\circ}\), \(60^{\circ}\) and \(90^{\circ}\), then the sides are in the ratio \(x:x\sqrt{3}:2x\).

The shorter leg is always x, the longer leg is always \(x\sqrt{3}\), and the hypotenuse is always \(2x\). If you ever forget these theorems, you can still use the Pythagorean Theorem.

What if you were given a 30-60-90 right triangle and the length of one of its side? How could you figure out the lengths of its other sides?

Example \(\PageIndex{1}\)

Find the value of \(x\) and \(y\).

f-d_f253e0915a3a06ec0d626be4e6d23bcd759bdf5480908ad2fcd50348+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{1}\)

Solution

We are given the longer leg.

\(\begin{aligned} &x\sqrt{3} =12 \\ &x=\dfrac{12}{\sqrt{3}}\cdot \dfrac{\sqrt{3}}{\sqrt{3}}=\dfrac{12\sqrt{3}}{3}=4\sqrt{3} \\ &\text{The hypotenuse is} \\ &y=2(4\sqrt{3})=8\sqrt{3}\end{aligned}\)

Example \(\PageIndex{2}\)

Find the value of \(x\) and \(y\).

f-d_54966f54aaec43bfd1b984a46d9b523714d601e30028766637c91a37+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{2}\)

Solution

We are given the hypotenuse.

\(\begin{aligned}&2x=16 \\ &x=8 \\ &\text{The longer leg is} \\ &y=8\cdot \sqrt{3}=8\sqrt{3}\end{aligned}\)

Example \(\PageIndex{3}\)

Find the length of the missing sides.

f-d_6cf56285ac6d4e253c51867d19ce5f1cf4dd5eb5ccd2c207bc43f65d+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{3}\)

Solution

We are given the shorter leg. If \(x=5\), then the longer leg, \(b=5\sqrt{3}\), and the hypotenuse, \(c=2(5)=10\).

Example \(\PageIndex{4}\)

Find the length of the missing sides.

f-d_382c5686a01a4042d0dc31e15fa8fa9f0185f564b927cd65b93dc128+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{4}\)

Solution

We are given the hypotenuse. \(2x=20\), so the shorter leg, \(f=\dfrac{20}{2}=10\), and the longer leg, \(g=10\sqrt{3}\).

Example \(\PageIndex{5}\)

A rectangle has sides 4 and \(4\sqrt{3}\). What is the length of the diagonal?

f-d_bfba1fae7bcbddafcff19d9ed2213028dcb883213bc48f78ea211ab7+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{5}\)

Solution

The two lengths are \(x\), \(x\sqrt{3}\), so the diagonal would be \(2x\), or \(2(4)=8\).

If you did not recognize this is a 30-60-90 triangle, you can use the Pythagorean Theorem too.

\(\begin{aligned} 4^2+(4\sqrt{3})^2&=d^2 \\ 16+48&=d^2 \\ d&=\sqrt{64}=8\end{aligned}\)

Review

In a 30-60-90 triangle, if the shorter leg is 5, then the longer leg is __________ and the hypotenuse is ___________.
In a 30-60-90 triangle, if the shorter leg is x, then the longer leg is __________ and the hypotenuse is ___________.
A rectangle has sides of length 6 and \(6\sqrt{3}\). What is the length of the diagonal?
Two (opposite) sides of a rectangle are 10 and the diagonal is 20. What is the length of the other two sides?

For questions 5-12, find the lengths of the missing sides. Simplify all radicals.

Figure \(\PageIndex{6}\)
Figure \(\PageIndex{7}\)
Figure \(\PageIndex{8}\)
Figure \(\PageIndex{9}\)
Figure \(\PageIndex{10}\)
Figure \(\PageIndex{11}\)
Figure \(\PageIndex{12}\)
Figure \(\PageIndex{13}\)

Review (Answers)

To see the Review answers, open this PDF file and look for section 8.6.

Resources

Vocabulary

Term	Definition
30-60-90 Theorem	If a triangle has angle measures of 30, 60, and 90 degrees, then the sides are in the ratio \(x : x \sqrt{3} : 2x\)
30-60-90 Triangle	A 30-60-90 triangle is a special right triangle with angles of \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\).
Hypotenuse	The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.
Legs of a Right Triangle	The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle.
Pythagorean Theorem	The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by \(a^2+b^2=c^2\), where a and b are legs of the triangle and c is the hypotenuse of the triangle.
Radical	The \(\sqrt\), or square root, sign.

Additional Resources

Interactive Element

Video: Solving Special Right Triangles

Activities: 30-60-90 Right Triangles Discussion Questions

Study Aids: Special Right Triangles Study Guide

Practice: 30-60-90 Right Triangles

Real World: Fighting the War on Drugs Using Geometry and Special Triangles