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1.10: 30-60-90 Right Triangles

  • Page ID
    14089
  • 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.

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

    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?

    Solution

    If you are not given a picture, draw one.

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

    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.


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