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How to Use the Pythagorean Theorem to Solve Lesson 6 Homework Problems

If you are struggling with lesson 6 homework practice on using the Pythagorean theorem, don’t worry. In this article, we will explain what the Pythagorean theorem is, how to use it, and how to check your answers with a key. By the end of this article, you will be able to solve any right triangle problem with confidence.

What is the Pythagorean Theorem?

The Pythagorean theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, often called the Pythagorean equation:

a + b = c

The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proven numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.

How to Use the Pythagorean Theorem?

The Pythagorean theorem can be used to find any missing side length of a right triangle, as long as you know two other side lengths. To use the theorem, you need to follow these steps:

1. Identify which side is the hypotenuse and which sides are the legs of the right triangle.
2. Plug in the known side lengths into the Pythagorean equation and solve for the unknown side length.
3. Simplify your answer and write it with the appropriate units.

Let’s look at an example:

In this right triangle, we know that a = 3 cm and b = 4 cm. We want to find c, which is the hypotenuse.

1. We plug in the known values into the Pythagorean equation: a + b = c
2. We get: 3 + 4 = c
3. We simplify: 9 + 16 = c
4. We get: 25 = c
5. We take the square root of both sides: c = √25
6. We simplify: c = 5
7. We write our answer with units: c = 5 cm

The Pythagorean theorem can also be used to check if a triangle is a right triangle or not. If a triangle has sides with lengths that satisfy the Pythagorean equation, then it must be a right triangle. Conversely, if a triangle does not have sides with lengths that satisfy the Pythagorean equation, then it cannot be a right triangle.

How to Check Your Answers with a Key?

To check your answers with a key, you need to compare your solutions with the correct ones provided by the key. You can do this by:

• Making sure you have used the correct formula and followed the correct steps.
• Making sure you have simplified your answers and written them with correct units.
• Making sure you have rounded your answers to the appropriate decimal places or fractions if needed.
• Making sure you have labeled your answers clearly and correctly.

If your answers match with the key, then you have done well. If your answers do not match with the key, then you need to review your work and find where you made a mistake. You can also ask for help from your teacher or peers if you are stuck.

How to Solve Lesson 6 Homework Practice Problems?

Now that you know how to use the Pythagorean theorem and check your answers with a key, you are ready to tackle some lesson 6 homework practice problems. Here are some tips to help you:

• Read the problem carefully and identify what you are given and what you need to find.
• Draw a diagram of the right triangle and label the sides with the given information.
• Use the Pythagorean theorem to write an equation and solve for the unknown side length.
• Check your answer by plugging it back into the equation and making sure it is true.
• Compare your answer with the key and see if it matches.

Let’s look at some examples:

Example 1: A ladder leans against a building. The base of the ladder is 8 feet from the building and the top of the ladder is 15 feet above the ground. How long is the ladder?

We are given that a = 8 ft and b = 15 ft. We need to find c, which is the length of the ladder.

1. We use the Pythagorean theorem to write an equation: a + b = c
2. We plug in the given values: 8 + 15 = c
3. We simplify: 64 + 225 = c
4. We get: 289 = c
5. We take the square root of both sides: c = √289
6. We simplify: c = 17
7. We write our answer with units: c = 17 ft

We check our answer by plugging it back into the equation: 8 + 15 = 17

We simplify: 64 + 225 = 289

We get: 289 = 289, which is true.

We compare our answer with the key and see that it matches.

Example 2: A rectangular rug has a length of 10 feet and a diagonal of 13 feet. What is the width of the rug?

We are given that b = 10 ft and c = 13 ft. We need to find a, which is the width of the rug.

1. We use the Pythagorean theorem to write an equation: a + b = c
2. We plug in the given values: a + 10 = 13
3. We simplify: a + 100 = 169
4. We subtract 100 from both sides: a = 69
5. We take the square root of both sides: a = √69
6. We approximate: a ≈ 8.3
7. We write our answer with units: a ≈ 8.3 ft

We check our answer by plugging it back into the equation: (8.3) + 10 ≈ 13

We simplify: 68.9 + 100 ≈ 169

We get: 168.9 ≈ 169, which is close enough.

We compare our answer with the key and see that it matches.

How to Find More Practice Problems and Answer Keys?

If you want to practice more problems on using the Pythagorean theorem, you can find many online resources that offer worksheets, quizzes, games, and videos on this topic. Some of these resources are:

• Quizlet: This website provides free solutions for Glencoe MATH Course 3, Volume 2, which includes lesson 6 on the Pythagorean theorem. You can also create your own flashcards and study sets on this topic.
• Khan Academy: This website offers free videos, exercises, and quizzes on the Pythagorean theorem and its applications. You can also track your progress and earn badges as you learn.
• ChiliMath: This website provides free worksheets and answer keys on the Pythagorean theorem practice problems. You can also find detailed explanations and examples on how to solve them.

These are just some of the online resources that you can use to practice and improve your skills on using the Pythagorean theorem. You can also ask your teacher or peers for more help if you need it.

How to Apply the Pythagorean Theorem to Real-World Problems?

The Pythagorean theorem is not only useful for solving geometry problems, but also for understanding and modeling many real-world situations. For example, you can use the Pythagorean theorem to:

• Find the distance between two points on a map or a coordinate plane.
• Find the height of a building, a tree, a mountain, or any other object by using its shadow and the angle of elevation.
• Find the diagonal of a screen, a box, a room, or any other rectangular object.
• Find the length of a ramp, a ladder, a slide, or any other inclined plane.
• Find the missing side of a right triangle in trigonometry, physics, engineering, or any other field that involves angles and forces.

Let’s look at some examples:

Example 3: A soccer field is 100 meters long and 60 meters wide. How far does a player run when he goes from one corner to the opposite corner of the field?

We are given that a = 100 m and b = 60 m. We need to find c, which is the distance that the player runs.

1. We use the Pythagorean theorem to write an equation: a + b = c
2. We plug in the given values: 100 + 60 = c
3. We simplify: 10000 + 3600 = c
4. We get: 13600 = c
5. We take the square root of both sides: c = √13600
6. We approximate: c ≈ 116.6
7. We write our answer with units: c ≈ 116.6 m

Example 4: A person is flying a kite that is attached to a string of length 50 feet. The string makes an angle of 40 degrees with the ground. How high is the kite above the ground?

We are given that c = 50 ft and ∠A = 40°. We need to find b, which is the height of the kite.

1. We use the Pythagorean theorem to write an equation: a + b = c
2. We use trigonometry to find a in terms of c and ∠A: a = c cos ∠A
3. We plug in the given values: (50 cos 40°) + b = 50
4. We simplify: (38.3) + b = 2500
5. We subtract (38.3) from both sides: b = 1054.5
6. We take the square root of both sides: b = √1054.5
7. We approximate: b ≈ 32.5
8. We write our answer with units: b ≈ 32.5 ft

Conclusion

In this article, we have learned how to use the Pythagorean theorem to solve lesson 6 homework practice problems. We have also learned how to check our answers with a key and how to apply the Pythagorean theorem to real-world problems. The Pythagorean theorem is a powerful and versatile tool that can help us understand and model many situations involving right triangles. We hope that this article has helped you master the Pythagorean theorem and improve your math skills.

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