1.2: Special Right Triangles (2024)

  1. Last updated
  2. Save as PDF
  • Page ID
    61218
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

    \( \newcommand{\vectorC}[1]{\textbf{#1}}\)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}}\)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}\)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)

    Learning Objectives

    • Recognize Special Right Triangles.
    • Use the special right triangle rations to solve special right triangles.

    30-60-90 Right Triangles

    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 \(y\).

    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 \(y\).

    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.

    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.

    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.

    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

    45-45-90 Right Triangles

    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).

    1.2: Special Right Triangles (6)

    \(\Delta ABC\) 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 \(\PageIndex{6}\)

    Find the length of \(x\).

    1.2: Special Right Triangles (8)

    Solution

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

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

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

    Example \(\PageIndex{7}\)

    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 \(\PageIndex{8}\)

    Find the length of the missing side.

    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 \(\PageIndex{9}\)

    Find the length of the missing side.

    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 \(\PageIndex{10}\)

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

    Solution

    Draw a picture.

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

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

    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?

    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 \(x : x \sqrt{3} : 2x\)
    45-45-90 Theorem For any isosceles right triangle, if the legs are x units long, the hypotenuse is always \(x\sqrt{2}\).
    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 45-45-90 Right Triangles

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

    1.2: Special Right Triangles (2024)

    FAQs

    Is 0.9 1.2 and 1.5 a right triangle? ›

    Only (0.9,1.2,1.5) is a right angled trangle.

    How do you find the answer to a special right triangle? ›

    Special Right Triangle Formula

    The special right triangle formulas in the form of ratios can be expressed as: 30° 60° 90° triangle formula: Short leg: Long leg : Hypotenuse = x: x√3: 2x. 45° 45° 90° triangle formula: Leg : Leg: Hypotenuse = x: x: x√2.

    What is a special right triangle 1 2 3? ›

    30° - 60° - 90° triangle

    This is a triangle whose three angles are in the ratio 1 : 2 : 3 and respectively measure 30° ( π/6), 60° ( π/3), and 90° ( π/2). The sides are in the ratio 1 : √3 : 2.

    Is 1.5 2 2.5 a right triangle? ›

    The sides of a triangle are of length 1.5 cm, 2.5 cm and 2 cm. Then this triangle is a right triangle.

    How to solve for right triangles? ›

    Solving right triangles

    We can use the Pythagorean theorem and properties of sines, cosines, and tangents to solve the triangle, that is, to find unknown parts in terms of known parts. Pythagorean theorem: a2 + b2 = c2. Sines: sin A = a/c, sin B = b/c. Cosines: cos A = b/c, cos B = a/c.

    What is the rule for the special right triangle? ›

    A 45-45-90 triangle is a special type of right triangle, where the ratio of the lengths of the sides of a 45-45-90 triangle is always 1:1:√2, meaning that if one leg is x units long, then the other leg is also x units long, and the hypotenuse is x√2 units long.

    How to remember 30-60-90 triangles? ›

    In any 30-60-90 triangle, you see the following: The shortest leg is across from the 30-degree angle, the length of the hypotenuse is always double the length of the shortest leg, and you can find the length of the long leg by multiplying the short leg by the square root of 3.

    Does 9 12 15 make a right triangle? ›

    Yes, 9, 12 and 15 is a Pythagorean Triple and sides of a right triangle. What are fractal patterns?

    Does 8, 10, and 12 make a right triangle? ›

    Answer and Explanation:

    Given a triangle having side lengths 8,10 and 12. As all the lengths are different from each other, this is a scalene triangle.

    Does 8, 15, and 17 make a right triangle? ›

    Determine if the following lengths are Pythagorean Triples. Plug the given numbers into the Pythagorean Theorem. Yes, 8, 15, 17 is a Pythagorean Triple and sides of a right triangle.

    Does 5, 12, and 13 make a right triangle? ›

    Does 5 12 and 13 make a right triangle? Yes, 5 12 and 13 make a right triangle. They are referred to as Pythagorean triplets, where 5 squared and 12 squared equal 13 squared, which is the application of the Pythagorean theorem.

    References

    Top Articles
    Latest Posts
    Article information

    Author: Golda Nolan II

    Last Updated:

    Views: 5981

    Rating: 4.8 / 5 (58 voted)

    Reviews: 89% of readers found this page helpful

    Author information

    Name: Golda Nolan II

    Birthday: 1998-05-14

    Address: Suite 369 9754 Roberts Pines, West Benitaburgh, NM 69180-7958

    Phone: +522993866487

    Job: Sales Executive

    Hobby: Worldbuilding, Shopping, Quilting, Cooking, Homebrewing, Leather crafting, Pet

    Introduction: My name is Golda Nolan II, I am a thoughtful, clever, cute, jolly, brave, powerful, splendid person who loves writing and wants to share my knowledge and understanding with you.