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4.42: 45-45-90 Right Triangles

  • Page ID
    4983
  • Leg times \(\sqrt{2}\) equals hypotenuse.

    A right triangle with congruent legs and acute angles is an Isosceles Right Triangle. This triangle is also called a 45-45-90 triangle (named after the angle measures).

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    Figure \(\PageIndex{1}\)

    \(\Delta ABC\) is a right triangle with \(m\angle A=90^{\circ}\), \(\overline{AB}\cong \overline{AC}\) and \(m\angle B=m\angle C=45^{\circ}\).

    45-45-90 Theorem: If a right triangle is isosceles, then its sides are in the ratio \(x:x:x\sqrt{2}\). For any isosceles right triangle, the legs are x and the hypotenuse is always \(x\sqrt{2}\).

    What if you were given an isosceles right triangle and the length of one of its sides? How could you figure out the lengths of its other sides?

    Example \(\PageIndex{1}\)

    Find the length of \(x\).

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    Figure \(\PageIndex{2}\)

    Solution

    Use the \(x:x:x\sqrt{2}\) ratio.

    Here, we are given the hypotenuse. Solve for \(x\) in the ratio.

    \(\begin{aligned} x\sqrt{2}&=16 \\ x&=\dfrac{16}{\sqrt{2}}\cdot \dfrac{\sqrt{2}}{\sqrt{2}}=\dfrac{16\sqrt{2}}{2}=8\sqrt{2}\end{aligned}\)

    Example \(\PageIndex{2}\)

    Find the length of \(x\), where \(x\) is the hypotenuse of a 45-45-90 triangle with leg lengths of \(5\sqrt{3}\).

    Solution

    Use the \(x:x:x\sqrt{2}\) ratio.

    \(x=5\sqrt{3}\cdot \sqrt{2}=5\sqrt{6}\)

    Example \(\PageIndex{3}\)

    Find the length of the missing side.

    f-d_eaae0d6f4193383b52eea952d89ad7d0326e67ed47b13b2f2d971c16+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    Solution

    Use the \(x:x:x\sqrt{2}\) ratio. \(TV=6\) because it is equal to \(ST\). So, \(SV=6\cdot \sqrt{2}=6\sqrt{2}\).

    Example \(\PageIndex{4}\)

    Find the length of the missing side.

    f-d_3d3f24ccca5492b908cd782520e20675d3ede898c5ec33f64a71392b+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    Solution

    Use the \(x:x:x\sqrt{2}\) ratio. \(AB=9\sqrt{2}\) because it is equal to \(AC\). So, \(BC=9\sqrt{2}\cdot \sqrt{2}=9\cdot 2=18\).

    Example \(\PageIndex{5}\)

    A square has a diagonal with length 10, what are the lengths of the sides?

    Solution

    f-d_5fc04f48c4f69c2d691f74e4f651c0f490d14c27053c327d6a10ac8c+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)

    We know half of a square is a 45-45-90 triangle, so \(10=s\sqrt{2}\).

    \(\begin{aligned} s\sqrt{2}&=10 \\ s&=10\sqrt{2}\cdot \sqrt{2}\sqrt{2}=10\sqrt{2}2=5\sqrt{2}\end{aligned}\)

    Review

    1. In an isosceles right triangle, if a leg is 4, then the hypotenuse is __________.
    2. In an isosceles right triangle, if a leg is \(x\), then the hypotenuse is __________.
    3. A square has sides of length 15. What is the length of the diagonal?
    4. A square’s diagonal is 22. What is the length of each side?

    For questions 5-11, find the lengths of the missing sides. Simplify all radicals.

    1. f-d_ad4ff816a423d2518b3e145ecab0af844148a342cf715f2822839d05+IMAGE_TINY+IMAGE_TINY.png
      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 \(\PageIndex{12}\)

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 8.5.

    Resources

    Vocabulary

    Term Definition
    45-45-90 Theorem For any isosceles right triangle, if the legs are x units long, the hypotenuse is always \(x\sqrt{2}\).
    45-45-90 Triangle A 45-45-90 triangle is a special right triangle with angles of \(45^{\circ}\), \(45^{\circ}\), 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.
    Radical The \(\sqrt\), or square root, sign.

    Additional Resources

    Interactive Element

    Video: Solving Special Right Triangles

    Activities: 45-45-90 Right Triangles Discussion Questions

    Study Aids: Special Right Triangles Study Guide

    Practice: 45-45-90 Right Triangles

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