4.43: 30-60-90 Right Triangles
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Hypotenuse equals twice the smallest leg, while the larger leg is √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∘, 60∘ and 90∘, then the sides are in the ratio x:x√3:2x.
The shorter leg is always x, the longer leg is always x√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 4.43.1
Find the value of x and y.

Solution
We are given the longer leg.
x√3=12x=12√3⋅√3√3=12√33=4√3The hypotenuse isy=2(4√3)=8√3
Example 4.43.2
Find the value of x and y.

Solution
We are given the hypotenuse.
2x=16x=8The longer leg isy=8⋅√3=8√3
Example 4.43.3
Find the length of the missing sides.

Solution
We are given the shorter leg. If x=5, then the longer leg, b=5√3, and the hypotenuse, c=2(5)=10.
Example 4.43.4
Find the length of the missing sides.

Solution
We are given the hypotenuse. 2x=20, so the shorter leg, f=202=10, and the longer leg, g=10√3.
Example 4.43.5
A rectangle has sides 4 and 4√3. What is the length of the diagonal?

Solution
The two lengths are x, x√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.
42+(4√3)2=d216+48=d2d=√64=8
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√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.
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Figure 4.43.6 -
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Figure 4.43.10 -
Figure 4.43.11 -
Figure 4.43.12 -
Figure 4.43.13
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√3:2x |
30-60-90 Triangle | A 30-60-90 triangle is a special right triangle with angles of 30∘, 60∘, and 90∘. |
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 a2+b2=c2, where a and b are legs of the triangle and c is the hypotenuse of the triangle. |
Radical | The \boldsymbol{\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