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45-45-90 Right Triangles

Leg times sqrt(2) equals hypotenuse.

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45-45-90 Right Triangles

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).

\begin{align*}\triangle ABC\end{align*}ABC is a right triangle with \begin{align*}m \angle A = 90^\circ\end{align*}mA=90, \begin{align*} \overline {AB} \cong \overline{AC}\end{align*}AB¯¯¯¯¯¯¯¯AC¯¯¯¯¯¯¯¯ and \begin{align*}m \angle B = m \angle C = 45^\circ\end{align*}mB=mC=45.

45-45-90 Theorem: If a right triangle is isosceles, then its sides are in the ratio \begin{align*}x:x:x \sqrt{2}\end{align*}x:x:x2. For any isosceles right triangle, the legs are \begin{align*}x\end{align*}x and the hypotenuse is always \begin{align*}x \sqrt{2}\end{align*}x2.

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?

Examples

Example 1

Find the length of \begin{align*}x\end{align*}x.

Use the \begin{align*}x:x:x \sqrt{2}\end{align*}x:x:x2 ratio.

Here, we are given the hypotenuse. Solve for \begin{align*}x\end{align*}x in the ratio.

\begin{align*}x \sqrt{2} &= 16\\ x &= \frac{16}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{16 \sqrt{2}}{2} = 8 \sqrt{2}\end{align*}x2x=16=16222=1622=82

Example 2

Find the length of \begin{align*}x\end{align*}x, where \begin{align*}x\end{align*}x is the hypotenuse of a 45-45-90 triangle with leg lengths of \begin{align*}5\sqrt{3}\end{align*}53.

Use the \begin{align*}x:x:x \sqrt{2}\end{align*}x:x:x2 ratio.

\begin{align*}x=5\sqrt{3}\cdot \sqrt{2}=5\sqrt{6}\end{align*}x=532=56

Example 3

Find the length of the missing side.

Use the \begin{align*}x:x:x \sqrt{2}\end{align*}x:x:x2 ratio. \begin{align*}TV = 6\end{align*}TV=6 because it is equal to \begin{align*}ST\end{align*}ST. So, \begin{align*}SV = 6 \cdot \sqrt{2} = 6 \sqrt{2}\end{align*}SV=62=62.

Example 4

Find the length of the missing side.

Use the \begin{align*}x:x:x \sqrt{2}\end{align*}x:x:x2 ratio. \begin{align*}AB = 9 \sqrt{2}\end{align*}AB=92 because it is equal to \begin{align*}AC\end{align*}AC. So, \begin{align*}BC = 9 \sqrt{2} \cdot \sqrt{2} = 9 \cdot 2 = 18\end{align*}BC=922=92=18.

Example 5

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

Draw a picture.

We know half of a square is a 45-45-90 triangle, so \begin{align*}10=s \sqrt{2}\end{align*}10=s2.

\begin{align*}s \sqrt{2} &= 10\\ s &= \frac{10}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}= \frac{10 \sqrt{2}}{2}=5 \sqrt{2}\end{align*}s2s=10=10222=1022=52

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 \begin{align*}x\end{align*}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.

Review (Answers)

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

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Vocabulary

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.

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