# 11.10: Applications Using the Pythagorean Theorem

**At Grade**Created by: CK-12

**Practice**Applications Using the Pythagorean Theorem

Suppose a you needed to rescue a cat from a tree. The cat is 40 feet up from the ground, and you place a ladder 30 feet away from the base of the tree. How tall does the ladder need to be to reach the cat? After completing this Concept, you'll be able to solve real-world applications like this one using the Pythagorean Theorem and its converse.

### Guidance

The Pythagorean Theorem and its converse have many applications for finding lengths and distances.

#### Example A

*Maria has a rectangular cookie sheet that measures 10 inches×14 inches. Find the length of the diagonal of the cookie sheet.*

**Solution**

**Draw a sketch:**

**Define variables:** Let

**Write a formula:** Use the Pythagorean Theorem:

**Solve the equation:**

**Check:**

#### Example B

*Find the area of the shaded region in the following diagram:*

**Solution**

Draw the diagonal of the square in the figure:

Notice that the diagonal of the square is also the diameter of the circle.

**Define variables:** Let

**Write the formula:** Use the Pythagorean Theorem:

**Solve the equation:**

The diameter of the circle is

Area of a circle formula:

The area of the shaded region is therefore

#### Example C

*In a right triangle, one leg is twice as long as the other and the perimeter is 28. What are the measures of the sides of the triangle?*

**Solution**

**Make a sketch and define variables:**

Let:

**Write formulas:**

The sides of the triangle are related in two different ways.

The perimeter is 28, so

The triangle is a right triangle, so the measures of the sides must satisfy the Pythagorean Theorem:

**Solve the equation:**

Plug the value of

The short leg is:

The long leg is:

The hypotenuse is:

**Check:** The legs of the triangle should satisfy the Pythagorean Theorem:

The perimeter of the triangle should be 28:

**The answer checks out.**

Watch this video for help with the examples above:

CK-12 Foundation: Applications Using the Pythagorean Theorem

### Guided Practice

*Mike is loading a moving van by walking up a ramp. The ramp is 10 feet long and the bed of the van is 2.5 feet above the ground. How far does the ramp extend past the back of the van?*

**Solution**

**Make a sketch:**

**Define variables:** Let

**Write a formula:** Use the Pythagorean Theorem:

**Solve the equation:**

**Check** by plugging the result in the Pythagorean Theorem:

**The answer checks out.**

### Explore More

- In order to make a ramp that is
3ft high and covers4ft of ground, how long must the ramp be? - A regulation baseball diamond is a square with 90 feet between bases. How far is second base from home plate?
- Emanuel has a cardboard box that measures
20 cm long × 10 cm wide × 8 cm deep .- What is the length of the diagonal across the bottom of the box?
- What is the length of the diagonal from a bottom corner to the opposite top corner?

- Samuel places a ladder against his house. The base of the ladder is 6 feet from the house and the ladder is 10 feet long.
- How high above the ground does the ladder touch the wall of the house?
- If the edge of the roof is 10 feet off the ground and sticks out 1.5 feet beyond the wall, how far is it from the edge of the roof to the top of the ladder?

- Find the area of the triangle below if the area of a triangle is defined as
A=12 base×height : - Instead of walking along the two sides of a rectangular field, Mario decided to cut across the diagonal. He thus saves a distance that is half of the long side of the field.
- Find the length of the long side of the field given that the short side is 123 feet.
- Find the length of the diagonal.

- Marcus sails due north and Sandra sails due east from the same starting point. In two hours Marcus’ boat is 35 miles from the starting point and Sandra’s boat is 28 miles from the starting point.
- How far are the boats from each other?
- Sandra then sails 21 miles due north while Marcus stays put. How far is Sandra from the original starting point?
- How far is Sandra from Marcus now?

- Determine the area of the circle below. (Hint: the hypotenuse of the triangle is the diameter of the circle.)
- A rectangle's length is
1in longer than its width and the diagonal has a length of29in . What are the lengths of the sides of the rectangle? - For an isosceles triangle with sides of the length given, find the length of each hypotenuse:
- 1
- 2
- 3
n

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 11.10.

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### Image Attributions

Here you'll learn how to apply the Pythagorean Theorem and its converse to solve real-world problems.

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