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We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.

Δ A B C and Δ X Y Z are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.

Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 3.2, and the side across from C is labeled 4. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4.5, the side across from Y is labeled y, and the side across from Z is labeled 3.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information. The figure is provided.
Step 2. Identify what you are looking for. The length of the sides of similar triangles
Step 3. Name. Choose a variable to represent it. Let
a = length of the third side of Δ A B C
y = length of the third side Δ X Y Z
Step 4. Translate.
The triangles are similar, so the corresponding sides are in the same ratio. So
A B X Y = B C Y Z = A C X Z

Since the side A B = 4 corresponds to the side X Y = 3 , we will use the ratio AB XY = 4 3 to find the other sides.

Be careful to match up corresponding sides correctly.
.
Step 5. Solve the equation. .
Step 6. Check:
.
Step 7. Answer the question. The third side of Δ A B C is 6 and the third side of Δ X Y Z is 2.4.
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Δ A B C is similar to Δ X Y Z . Find a .

Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.

8

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Δ A B C is similar to Δ X Y Z . Find y .

Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.

22.5

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Use the pythagorean theorem

The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around 500 BCE.

Remember that a right triangle has a 90° angle, which we usually mark with a small square in the corner. The side of the triangle opposite the 90° angle is called the hypotenuse    , and the other two sides are called the legs . See [link] .

Three right triangles are shown. Each has a box representing the right angle. The first one has the right angle in the lower left corner, the next in the upper left corner, and the last one at the top. The two sides touching the right angle are labeled “leg” in each triangle. The sides across from the right angles are labeled “hypotenuse.”
In a right triangle, the side opposite the 90° angle is called the hypotenuse and each of the other sides is called a leg.

The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.

The pythagorean theorem

In any right triangle Δ A B C ,

a 2 + b 2 = c 2

where c is the length of the hypotenuse a and b are the lengths of the legs.

A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.

To solve problems that use the Pythagorean Theorem, we will need to find square roots. In Simplify and Use Square Roots we introduced the notation m and defined it in this way:

If m = n 2 , then m = n for n 0

For example, we found that 25 is 5 because 5 2 = 25 .

We will use this definition of square roots to solve for the length of a side in a right triangle.

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

Right triangle with legs labeled as 3 and 4.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for. the length of the hypotenuse of the triangle
Step 3. Name. Choose a variable to represent it. Let c = the length of the hypotenuse
.
Step 4. Translate.
Write the appropriate formula.
Substitute.

.
Step 5. Solve the equation. .
Step 6. Check:
.
Step 7. Answer the question. The length of the hypotenuse is 5.
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Use the Pythagorean Theorem to find the length of the hypotenuse.

A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.

10

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

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as c. One of the sides touching the right angle is labeled as 15, the other is labeled “8”.

17

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Use the Pythagorean Theorem to find the length of the longer leg.

Right triangle is shown with one leg labeled as 5 and hypotenuse labeled as 13.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for. The length of the leg of the triangle
Step 3. Name. Choose a variable to represent it. Let b = the leg of the triangle
Label side b
.
Step 4. Translate.
Write the appropriate formula. Substitute.
.
Step 5. Solve the equation. Isolate the variable term. Use the definition of the square root.
Simplify.
.
Step 6. Check:
.
Step 7. Answer the question. The length of the leg is 12.
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Questions & Answers

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Practice Key Terms 9

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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