## Introduction

In this chapter, we will learn the application of the Pythagoras theorem to solve problems, the application of the Pythagoras theorem to a triangle in three dimensional figures and the application of converse of Pythagoras theorem to solve problems and to identify right angle triangles

### 7.3.1 Apply the Pythagorean Theorem to Solve Problems

**Example1:**

Pamela is placing a ladder of 41 m long on the ground in such a way that it touches the top of a vertical wall, which is 40 m high. She is wondering to find the distance of the foot of the ladder from the bottom of the wall. Can you help Pamela?

**Solution:**

Let the required distance be *x* meters. Here, the ladder, the wall and the ground form a right-angled triangle. The ladder is the hypotenuse of that triangle.

According to the Pythagorean theorem,

a^{2} + b^{2} = c^{2}

a^{2} + 40^{2} = 41^{2}

⇒ a^{2} = 41^{2} – 40^{2}

⇒ a^{2} = (41 + 40) (41 – 40)

⇒ a^{2} = (81) (1)

⇒ a^{2} = 81

⇒ a = √81

⇒ a = 9

Therefore, the distance of the foot of the ladder from the bottom of the wall = 5 meters.

**Example2:**

Patrick wants to go to the store from his home to buy a tennis racket. He can either take the sidewalk all the way or cut across the field at the corner. How much shorter is the trip if he cuts across the field?

**Solution:**

Let the required distance be *x* meters. Here, the shortest route, road 1 and road 2 form a right-angled triangle. The shortest route is the hypotenuse of that triangle.

The total length of road 1 is 20 + 70 = 90

The total length of road 2 is 50 + 70 = 120

According to the Pythagorean theorem,

a^{2} + b^{2} = c^{2}

90^{2} + 120^{2} = c^{2}

⇒ 8100 + 14400 = c^{2}

⇒ 22500 = c^{2}

⇒ √22500 = c

⇒ 150 = c

Therefore, the trip will be 150 meters if he cuts across the field.

### 7.3.2 Apply the Pythagorean Theorem to Triangles in Three Dimensions

**Example1:**

Carlos wants to present his friend with a new umbrella on his birthday. She has purchased one in the store.

She also made a box with the following dimensions to pack the umbrella.

She is wondering whether the umbrella will fit in the box or not? Can you help Carlos to decide whether she can use this box or not?

**Solution:**

**Step 1: **Find the length of the diagonal, *d*, of the bottom of the box.

22^{2} + 27^{2} = d^{2}

484 + 729 = d^{2}

1213 = d^{2}

√1213 = d

34.82 = d

**Step 2:** Use the Pythagorean theorem to find the hypotenuse of the box.

14^{2} + 34.82^{2} = c^{2}

196 + 1212.43 = c^{2}

1408.43 = c^{2}

√1408.43 = c

37.52 = c

Since the length of the umbrella is 37.5 inches, Carlos can use this box to pack the umbrella.

### 7.3.3 Apply the Converse of the Pythagorean Theorem to Solve Problems

**Example 1:**

Dennis is making triangles for a stained glass window. He made the design shown below but wants to change it. Dennis wants to move the purple tringle to the corner. The purple piece has side lengths of 4.5 inches, 6 inches and 7 inches. Can Dennis move the purple piece to the corner? Explain.

**Solution:**

Use the converse of the Pythagorean theorem to determine if the triangle is a right triangle.

a^{2} + b^{2} = c^{2}

4.5^{2} + 6^{2} = 7^{2}

20.5 + 36 = 49

56.25 = 49

Since 56.25 is not equal to 49, it is not a right angle triangle. Dennis cannot place this triangle in the corner.

## Exercise:

- Rebecca marched 6 m east and 12 m north. How far is he from his starting point?
- The rectangle PQRS represents the floor of a room.

Keith stands at point A. Carl stands at point S. Terry stands at point R.

a) Calculate the distance of Keith from Carl.

b) Calculate the distance of Keith from Terry.

- A 35-foot ladder is leaning against the side of a building and is positioned such that the base of the ladder is 21 feet from the base of the building. How far above the ground is the point where the ladder touches the building?
- A carpet measures 7 feet long and has a diagonal measurement of √74 feet. Find the width of the carpet.
- Find the missing lengths in the rectangular prism.

- A box-shaped like a right rectangular prism measures 7 centimeters by 5 centimeters by 2 centimeters. What is the length of the interior diagonal of the prism to the nearest hundredths?
- A stainless steel patio heater is shaped like a square pyramid. The length of one side of the base is 19.8 inches. The slant height is 92.8 inches. What is the height of the heater? Round to the nearest tenth of an inch.
- Jacqueline is buying edging for a triangular flower garden she plans to build in her backyard. If the lengths of the three pieces of edging that she purchases are 10 feet, 12 feet, and 6 feet, will the flower garden be in the shape of a right triangle?
- A triangular box has sides that measure 16 inches, 17 inches, and 24 inches. Is the box in the shape of a right triangle? Explain.

- A blueprint for a new triangular shape shows that the sides measure 18.5 ft, 6 ft, 17.5 ft. Is the playground in the shape of a right triangle? Explain

### What have we learned:

■ Application of Pythagoras theorem to solve problems.

■ Application of Pythagoras theorem to triangle in three dimensional figures.

■ Application of Converse of Pythagoras theorem to Solve Problems and to identify right angle triangles.

### Concept Map

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