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Because the Pythagorean Theorem contains variables that are squared, to solve for the length of a side in a right triangle, we will have to use square roots.

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

A right triangle with legs marked 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.
Label side c on the figure.
Let c = the length of the hypotenuse.

.
Step 4. Translate.
Write the appropriate formula. a 2 + b 2 = c 2
Substitute. 3 2 + 4 2 = c 2
Step 5. Solve the equation. 9 + 16 = c 2
Simplify. 25 = c 2
Use the definition of square root. 25 = c
Simplify. 5 = c
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 in the triangle shown below.

A right triangle with legs marked 6 and 8. The hypotenuse is marked c.

c = 10

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Use the Pythagorean Theorem to find the length of the hypotenuse in the triangle shown below.

No Alt Text

c = 13

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Use the Pythagorean Theorem to find the length of the leg shown below.

A right angle with one leg marked 5. The hypotenuse is labeled 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.
Lable side b . .
Step 4. Translate
Write the appropriate formula. a 2 + b 2 = c 2
Substitute. 5 2 + b 2 = 13 2
Step 5. Solve the equation. 25 + b 2 = 169
Isolate the variable term. b 2 = 144
Use the definition of square root. b 2 = 144
Simplify. b = 12
Step 6. Check.

.
Step 7. Answer the question. The length of the leg is 12.

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Use the Pythagorean Theorem to find the length of the leg in the triangle shown below.

A right triangle with legs marked b and 15. The hypotenuse is marked 17.

8

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Use the Pythagorean Theorem to find the length of the leg in the triangle shown below.

A right triangle with legs marked b and 9. The hypotenuse is marked 15.

12

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A gazebo is shown. In one of its corners, a triangle is made with the wood. The hypotenuse is marked 10 inches, and one of the legs is marked x

Kelvin is building a gazebo and wants to brace each corner by placing a 10 piece of wood diagonally as shown above.

If he fastens the wood so that the ends of the brace are the same distance from the corner, what is the length of the legs of the right triangle formed? Approximate to the nearest tenth of an inch.

Solution

Step 1. Read the problem. Step 2. Identify what we are looking for. the distance from the corner that the bracket should be attached Step 3. Name. Choose a variable to represent it. Let x = the distance from the corner. Step 4. Translate. Write the appropriate formula and substitute. Step 5. Solve the equation. Isolate the variable. Use the definition of square root. Simplify. Approximate to the nearest tenth. a 2 + b 2 = c 2 x 2 + x 2 = 10 2 2 x 2 = 100 x 2 = 50 x = 50 x 7.1 Step 6. Check. a 2 + b 2 = c 2 ( 7.1 ) 2 + ( 7.1 ) 2 10 2 Yes. Step 7. Answer the question. Kelvin should fasten each piece of wood approximately 7.1 from the corner.

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John puts the base of a 13-foot ladder five feet from the wall of his house as shown below. How far up the wall does the ladder reach?

A house is shown with a ladder leaning against it. The ladder is marked 13’, and the distance from the house to the base of the ladder is marked 5’.

12 feet

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Randy wants to attach a 17 foot string of lights to the top of the 15 foot mast of his sailboat, as shown below. How far from the base of the mast should he attach the end of the light string?

A sailboat is shown with a 15’ mast (the straight tall part). From the top of the mast, a series of colored dots stretches down to the back of the boat and is marked 17’.

8 feet

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Solve applications using rectangle properties

You may already be familiar with the properties of rectangles. Rectangles have four sides and four right ( 90 ° ) angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, L , and its adjacent side as the width, W .

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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