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Special Right Triangles

Properties of 30-60-90 and 45-45-90 triangles.

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Special Right Triangles

The Pythagorean Theorem is great for finding the third side of a right triangle when you already know two other sides. There are some triangles like 30-60-90 and 45-45-90 triangles that are so common that it is useful to know the side ratios without doing the Pythagorean Theorem each time. Using these patterns also allows you to totally solve for the missing sides of these special triangles when you only know one side length.

Given a 45-45-90 right triangle with sides 6 inches, 6 inches and x inches\begin{align*}x \ \text{inches}\end{align*}, what is the value of x\begin{align*}x\end{align*}

Guidance

A 30-60-90 right triangle has side ratios x,x3,2x\begin{align*}x, x \sqrt{3}, 2x\end{align*}

Confirm with Pythagorean Theorem:

x2+(x3)2x2+3x24x2=(2x)2=4x2=4x2\begin{align*}x^2 + \left ( x \sqrt{3} \right )^2 & = (2x)^2\\ x^2 + 3x^2 & = 4x^2\\ 4x^2 & = 4x^2\end{align*}

A 45-45-90 right triangle has side ratios x,x,x2\begin{align*}x, x, x \sqrt{2}\end{align*}

Confirm with Pythagorean Theorem:

x2+x22x2=(x2)2=2x2\begin{align*}x^2 + x^2 & = \left ( x \sqrt{2} \right )^2\\ 2x^2 & = 2x^2\end{align*}

Note that the order of the side ratios x,x3,2x\begin{align*}x, x \sqrt{3}, 2x\end{align*} and x,x,x2\begin{align*}x, x, x \sqrt{2}\end{align*} is important because each side ratio has a corresponding angle. In all triangles, the smallest sides correspond to smallest angles and largest sides always correspond to the largest angles.

Pythagorean number triples are special right triangles with integer sides. While the angles are not integers, the side ratios are very useful to know because they show up everywhere. Knowing these number triples also saves a lot of time from doing the Pythagorean Theorem repeatedly. Here are some examples of Pythagorean number triples:

• 3, 4, 5
• 5, 12, 13
• 7, 24, 25
• 8, 15, 17
• 9, 40, 41

More Pythagorean number triples can be found by scaling any other Pythagorean number triple. For example:

3,4,56,8,10\begin{align*}3, 4, 5 \rightarrow 6, 8, 10\end{align*} (scaled by a factor of 2)

Even more Pythagorean number triples can be found by taking any odd integer like 11, squaring it to get 121, halving the result to get 60.5. The original number 11 and the two numbers that are 0.5 above and below (60 and 61) will always be a Pythagorean number triple.

112+602=612\begin{align*}11^2 + 60^2 = 61^2\end{align*}

Example A

A right triangle has two sides that are 3 inches. What is the length of the third side?

Solution: Since it is a right triangle and it has two sides of equal length then it must be a 45-45-90 right triangle.  The third side is 32 inches\begin{align*}3 \sqrt{2} \ \text{inches}\end{align*}.

Example B

A 30-60-90 right triangle has hypotenuse of length 10. What are the lengths of the other two sides?

Solution: The hypotenuse is the side opposite 90. Sometimes it is helpful to draw a picture or make a table.

 30 60 90 x\begin{align*}x\end{align*} x3√\begin{align*}x \sqrt{3}\end{align*} 2x\begin{align*}2x\end{align*} 10

From the table you can write very small subsequent equations to solve for the missing sides.

10xx3=2x=5=53\begin{align*}10 & = 2x\\ x & = 5\\ x \sqrt{3} & = 5 \sqrt{3}\end{align*}

Example C

A 30-60-90 right triangle has a side length of 18 inches corresponding to 60 degrees. What are the lengths of the other two sides?

Solution: Make a table with the side ratios and the information given, then write equations and solve for the missing side lengths.

 30 60 90 x\begin{align*}x\end{align*} x3√\begin{align*}x \sqrt{3}\end{align*} 2x\begin{align*}2x\end{align*} 18

18183x=x3=x=183=18333=1833=63\begin{align*}18 & = x \sqrt{3}\\ \frac{18}{\sqrt{3}} & = x\\ x & = \frac{18}{\sqrt{3}} = \frac{18}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{18 \sqrt{3}}{3} = 6 \sqrt{3}\end{align*}

Note that you need to rationalize denominators.

Concept Problem Revisited

If you can recognize the pattern for 45-45-90 right triangles, a right triangle with legs 6 inches and 6 inches has a hypotenuse that  is 62 inches\begin{align*}6 \sqrt{2} \ \text{inches}\end{align*}.  x=62\begin{align*}x=6\sqrt{2}\end{align*}.

Vocabulary

Corresponding angles and sides are angles and sides that are on opposite sides of each other in a triangle. Capital letters like A,B,C\begin{align*}A, B, C\end{align*} are often used for the angles in a triangle and the lower case letters a,b,c\begin{align*}a, b, c\end{align*} are used for their corresponding sides (angle A\begin{align*}A\end{align*} corresponds to side a etc).

Pythagorean number triples are special right triangles with integer sides.

A 45-45-90 triangle is a special right triangle with angles of 45,45\begin{align*}45^\circ, 45^\circ\end{align*}, and 90\begin{align*}90^\circ\end{align*}.

A 30-60-90 triangle is a special right triangle with angles of 30,60\begin{align*}30^\circ, 60^\circ\end{align*}, and 90\begin{align*}90^\circ\end{align*}.

Guided Practice

Using your knowledge of special right triangle ratios, solve for the missing sides of the following right triangles.

1.

2.

3.

1. The other sides are each 522\begin{align*}\frac{5 \sqrt{2}}{2}\end{align*}.

 45 45 90 x\begin{align*}x\end{align*} x\begin{align*}x\end{align*} x2√\begin{align*}x \sqrt{2}\end{align*} 5

x2x=5=5222=522\begin{align*}x \sqrt{2} & = 5\\ x & = \frac{5}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{5 \sqrt{2}}{2}\end{align*}

2. The other sides are 2\begin{align*}\sqrt{2}\end{align*} and 6\begin{align*}\sqrt{6}\end{align*}.

 30 60 90 x\begin{align*}x\end{align*} x3√\begin{align*}x \sqrt{3}\end{align*} 2x\begin{align*}2x\end{align*} 22√\begin{align*}2 \sqrt{2}\end{align*}

2xxx3=22=2=23=6\begin{align*}2x & = 2 \sqrt{2}\\ x & = \sqrt{2}\\ x \sqrt{3} & = \sqrt{2} \cdot \sqrt{3} = \sqrt{6}\end{align*}

3. The other sides are 9 and 63\begin{align*}6 \sqrt{3}\end{align*}.

 30 60 90 x\begin{align*}x\end{align*} x3√\begin{align*}x \sqrt{3}\end{align*} 2x\begin{align*}2x\end{align*} 33√\begin{align*}3 \sqrt{3}\end{align*}

x2xx3=33=63=333=9\begin{align*}x & = 3 \sqrt{3}\\ 2x & = 6 \sqrt{3}\\ x \sqrt{3} & = 3 \sqrt{3} \cdot \sqrt{3} = 9\end{align*}

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