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3.11: Special Right Triangles, 45–45–90

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

  • Identify and use the ratios involved with right isosceles triangles.

Right Isosceles Triangles

What happens when you draw a diagonal across a square? Try it in the margin. \begin{align*}\rightarrow\end{align*}

You get two isosceles right triangles. Since a square has 4 right angles inside, 2 of them stay complete when you make the diagonal and the other 2 are cut in half. Each of the half angles are \begin{align*}45^\circ\end{align*} (because \begin{align*}90^\circ \div 2 = 45^\circ\end{align*}.)

  • Each triangle has the angles \begin{align*}45^\circ, 45^\circ\end{align*} (from the two angles cut in half), and ___________.

The diagonal becomes the hypotenuse of each isosceles right triangle because it is across from the right angle. Since a square has 4 congruent sides, each triangle is isosceles where the legs are the congruent sides of the square.

  • The diagonal of the square becomes the __________________________ of each triangle.

Each of these right triangles is a special right triangle called the \begin{align*}45^\circ - 45^\circ - 90^\circ\end{align*} right triangles (because the angles inside the triangle are \begin{align*}45^\circ, 45^\circ\end{align*}, and \begin{align*}90^\circ\end{align*}.)

As you know, isosceles triangles have two sides that are the same length. Additionally, the base angles of an isosceles triangle are congruent. An isosceles right triangle will always have base angles that each measure \begin{align*}45^\circ\end{align*} and a vertex angle that measures \begin{align*}90^\circ\end{align*}.

In the diagram above, the ______________ and the base ________________ are each congruent.

Don’t forget that the base angles are the angles that are opposite the congruent sides. They don’t have to be on the bottom of the figure, like in the picture below:

Example 1

The isosceles right triangle below has legs measuring 1 centimeter.

Use the Pythagorean Theorem to find the length of the hypotenuse.

Since the triangle is isosceles, the legs are 1 centimeter each. Substitute 1 for both \begin{align*}a\end{align*} and \begin{align*}b\end{align*}, and solve for \begin{align*}c\end{align*}:

\begin{align*}a^2+b^2 &= c^2\\ 1^2+1^2 &= c^2\\ 1+1 &= c^2\\ 2 &= c^2\\ \sqrt{2} &= \sqrt{c^2}\\ c & = \sqrt{2}\end{align*}

In this example, \begin{align*}c=\sqrt{2} \ cm\end{align*}.

What if each leg in the example above was 5 cm? Then we would have:

\begin{align*}a^2+b^2 &= c^2\\ 5^2+5^2 &= c^2\\ 25+25 &= c^2\\ 50 &= c^2\\ \sqrt{50} &= \sqrt{c^2}\\ c &= 5 \sqrt{2}\end{align*}

If each leg is 5 cm, then the hypotenuse is \begin{align*}5 \sqrt{2} \ cm\end{align*}.

When the length of each leg was 1, the hypotenuse was \begin{align*}1 \sqrt{2}\end{align*}.

When the length of each leg was 5, the hypotenuse was \begin{align*}5 \sqrt{2}\end{align*}.

Is this a coincidence? No. Recall that the legs of all \begin{align*}45^\circ - 45^\circ - 90^\circ\end{align*} triangles are proportional.

What does proportional mean?

You may recognize the word “proportion,” which means “ratio” or “fraction.”

“Proportional” describes a relationship between 2 values where you can multiply one of the values by some number and get the second value.

For instance, 3 and 6 have the same “proportional” relationship as 4 and 8, because you need to multiply the first number by 2 to get the second number in both cases.

Another pair of numbers with the same proportional relationship is ________ and _________.

Another example is the sentence: “Punishment should be proportional to the crime”.

This means that the worse a crime is, the harsher the punishment should be.

As we discovered in the examples on the previous page,

The hypotenuse of an isosceles right triangle will always equal the product of the length of one leg and \begin{align*}\sqrt{2}\end{align*}.

This means that if a leg has a length of \begin{align*}x\end{align*}, then you can multiply the leg by the number \begin{align*}\sqrt{2}\end{align*} to get the hypotenuse. So the hypotenuse has a length of \begin{align*}x \sqrt{2}\end{align*}.

In all \begin{align*}45^\circ - 45^\circ - 90^\circ\end{align*} triangles:

  • The length of the ___________________________ equals \begin{align*}\sqrt{2}\end{align*} times the length of a leg.

This relationship is very important to know! 

Example 2

What is the length of the hypotenuse in the triangle below?

We just learned a relationship between the leg and the hypotenuse of a \begin{align*}45^\circ - 45^\circ - 90^\circ\end{align*} triangle, so this problem is much easier than using the Pythagorean Theorem again like in Example 1!

First, we must determine which side of the triangle is the hypotenuse.

Since the hypotenuse is the longest _____________ and it is across from the _______________ angle, it must be side \begin{align*}c\end{align*}.

This makes the legs the other two sides, which have a length of _________________.

Since the length of the hypotenuse is the product of one leg and \begin{align*}\sqrt{2}\end{align*}, you can calculate this length \begin{align*}(c)\end{align*} by multiplying the leg by \begin{align*}\sqrt{2}\end{align*}.

One leg is 4 inches, so the hypotenuse \begin{align*}( c )\end{align*} will be \begin{align*}4 \sqrt{2}\end{align*} inches, or about 5.66 inches.

Reading Check:

1. Every isosceles right triangle has 3 special interior angles. What are they?

__________ , __________ , and __________

2. If an isosceles right triangle has legs that are 3 inches long, how long is its hypotenuse?

a. Draw a picture of the triangle here:




b. Use the Pythagorean Theorem to find the length of the hypotenuse (like in Example 1):




c. Use the special proportional relationship to find the length of the hypotenuse (like in Example 2):




d. Are your answers to (b.) and (c.) above the same?




Example 3

Antonio built a square patio in his backyard.

He wants to make a water pipe for flowers that goes from one corner to another, diagonally. How long will that pipe be?

The first step in a word problem like this is to add important information to the drawing. Because the problem asks you to find the length from one corner to another, you should draw a diagonal line segment (from one corner of the square to the opposite corner) into your patio picture:

Once you draw the diagonal path, you can see how triangles help answer this question.

Because both legs of the triangle have the same measurement (17 feet), this is an isosceles right triangle. The angles in an isosceles right triangle are \begin{align*}45^\circ, 45^\circ,\end{align*} and \begin{align*}90^\circ\end{align*}

In an isosceles right triangle, the hypotenuse is always equal to the product of the length of one leg and \begin{align*}\sqrt{2}\end{align*}. Just multiply these values together!

So, the length of Antonio’s water pipe will be the product of 17 and \begin{align*}\sqrt{2}\end{align*}, or \begin{align*}17\sqrt{2}\approx 17 \cdot (1.414) \ feet.\end{align*} This value is approximately equal to 24.04 feet. Therefore, his diagonal water pipe should be 24.04 feet long.

Reading Check:

You cook a grilled cheese sandwich. To make it easier to eat, you cut the sandwich in half diagonally. If each slice of bread (before it is cut) measures 14 cm by 14 cm, how long is the diagonal of your sandwich?

(Hint: draw yourself a picture to start this problem! If you are stuck, look at Example 3 to help you.)





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