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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

  1. In a 30-60-90 triangle, if the shorter leg is 5, then the longer leg is __________ and the hypotenuse is ___________.
  2. In a 30-60-90 triangle, if the shorter leg is x, then the longer leg is __________ and the hypotenuse is ___________.
  3. A rectangle has sides of length 6 and 6\sqrt{3}. What is the length of the diagonal?
  4. 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.

  1. f-d_2facd36bb2fbeedd849497dd107ba534533e6a88784cf8456c710384+IMAGE_TINY+IMAGE_TINY.png
    Figure \PageIndex{6}
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    Figure \PageIndex{7}
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    Figure \PageIndex{8}
  4. f-d_c4d28ed45dc897a6ead6481b0de495a862a1228b6a361f6912b49eef+IMAGE_TINY+IMAGE_TINY.png
    Figure \PageIndex{9}
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    Figure \PageIndex{10}
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    Figure \PageIndex{11}
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    Figure \PageIndex{12}
  8. f-d_6a40679372417fa67db43201a11e0a3f5cb60ec698a15303644d8570+IMAGE_TINY+IMAGE_TINY.png
    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


This page titled 4.43: 30-60-90 Right Triangles is shared under a CK-12 license and was authored, remixed, and/or curated by CK-12 Foundation via source content that was edited to the style and standards of the LibreTexts platform.

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