Pythagorean Theorem Worksheets - Free Printable
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Step-by-step solution for: Pythagorean Theorem Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Pythagorean Theorem Worksheets
It seems like you've uploaded an image containing a worksheet related to the Pythagorean Theorem. However, the image itself is not visible in this conversation. To help you solve the problems, I'll provide a general explanation of how to approach each type of problem typically associated with the Pythagorean Theorem. If you can describe the specific problems or tasks from the worksheet, I can tailor the solution accordingly.
The Pythagorean Theorem states that in a right triangle:
\[
a^2 + b^2 = c^2
\]
where:
- \(a\) and \(b\) are the lengths of the two legs (the sides forming the right angle),
- \(c\) is the length of the hypotenuse (the side opposite the right angle).
1. Finding the Hypotenuse:
- Given the lengths of the two legs (\(a\) and \(b\)), find the hypotenuse (\(c\)).
- Formula: \(c = \sqrt{a^2 + b^2}\).
2. Finding a Leg:
- Given the length of one leg and the hypotenuse, find the other leg.
- Formula for finding \(a\): \(a = \sqrt{c^2 - b^2}\).
- Formula for finding \(b\): \(b = \sqrt{c^2 - a^2}\).
3. Word Problems:
- Real-world scenarios where a right triangle is involved (e.g., ladders leaning against walls, diagonals of rectangles, etc.).
4. Verification:
- Checking if a triangle is a right triangle by verifying if \(a^2 + b^2 = c^2\).
---
#### Problem 1: Finding the Hypotenuse
Example: A right triangle has legs of lengths 6 cm and 8 cm. Find the length of the hypotenuse.
Solution:
Using the Pythagorean Theorem:
\[
c = \sqrt{a^2 + b^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm}
\]
#### Problem 2: Finding a Leg
Example: In a right triangle, the hypotenuse is 13 cm, and one leg is 5 cm. Find the length of the other leg.
Solution:
Using the Pythagorean Theorem:
\[
b = \sqrt{c^2 - a^2} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \text{ cm}
\]
#### Problem 3: Word Problem
Example: A ladder is leaning against a wall. The base of the ladder is 12 feet away from the wall, and the top of the ladder reaches 16 feet up the wall. How long is the ladder?
Solution:
Here, the ladder forms the hypotenuse of a right triangle, with the base of the ladder as one leg and the height it reaches on the wall as the other leg.
\[
c = \sqrt{a^2 + b^2} = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \text{ feet}
\]
#### Problem 4: Verification
Example: Determine if a triangle with sides 7 cm, 24 cm, and 25 cm is a right triangle.
Solution:
Check if \(a^2 + b^2 = c^2\):
\[
7^2 + 24^2 = 49 + 576 = 625
\]
\[
25^2 = 625
\]
Since \(7^2 + 24^2 = 25^2\), the triangle is a right triangle.
---
If you can describe the specific problems from your worksheet, I can provide detailed solutions tailored to those questions. For now, the general approach and examples above should help you solve most Pythagorean Theorem problems.
If you have any specific questions or need further clarification, feel free to ask!
Final Answer:
\boxed{\text{Provide the specific problems for a detailed solution.}}
General Explanation of the Pythagorean Theorem
The Pythagorean Theorem states that in a right triangle:
\[
a^2 + b^2 = c^2
\]
where:
- \(a\) and \(b\) are the lengths of the two legs (the sides forming the right angle),
- \(c\) is the length of the hypotenuse (the side opposite the right angle).
Common Types of Problems Related to the Pythagorean Theorem
1. Finding the Hypotenuse:
- Given the lengths of the two legs (\(a\) and \(b\)), find the hypotenuse (\(c\)).
- Formula: \(c = \sqrt{a^2 + b^2}\).
2. Finding a Leg:
- Given the length of one leg and the hypotenuse, find the other leg.
- Formula for finding \(a\): \(a = \sqrt{c^2 - b^2}\).
- Formula for finding \(b\): \(b = \sqrt{c^2 - a^2}\).
3. Word Problems:
- Real-world scenarios where a right triangle is involved (e.g., ladders leaning against walls, diagonals of rectangles, etc.).
4. Verification:
- Checking if a triangle is a right triangle by verifying if \(a^2 + b^2 = c^2\).
---
Example Solutions for Typical Problems
#### Problem 1: Finding the Hypotenuse
Example: A right triangle has legs of lengths 6 cm and 8 cm. Find the length of the hypotenuse.
Solution:
Using the Pythagorean Theorem:
\[
c = \sqrt{a^2 + b^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm}
\]
#### Problem 2: Finding a Leg
Example: In a right triangle, the hypotenuse is 13 cm, and one leg is 5 cm. Find the length of the other leg.
Solution:
Using the Pythagorean Theorem:
\[
b = \sqrt{c^2 - a^2} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \text{ cm}
\]
#### Problem 3: Word Problem
Example: A ladder is leaning against a wall. The base of the ladder is 12 feet away from the wall, and the top of the ladder reaches 16 feet up the wall. How long is the ladder?
Solution:
Here, the ladder forms the hypotenuse of a right triangle, with the base of the ladder as one leg and the height it reaches on the wall as the other leg.
\[
c = \sqrt{a^2 + b^2} = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \text{ feet}
\]
#### Problem 4: Verification
Example: Determine if a triangle with sides 7 cm, 24 cm, and 25 cm is a right triangle.
Solution:
Check if \(a^2 + b^2 = c^2\):
\[
7^2 + 24^2 = 49 + 576 = 625
\]
\[
25^2 = 625
\]
Since \(7^2 + 24^2 = 25^2\), the triangle is a right triangle.
---
Request for Specific Problems
If you can describe the specific problems from your worksheet, I can provide detailed solutions tailored to those questions. For now, the general approach and examples above should help you solve most Pythagorean Theorem problems.
If you have any specific questions or need further clarification, feel free to ask!
Final Answer:
\boxed{\text{Provide the specific problems for a detailed solution.}}
Parent Tip: Review the logic above to help your child master the concept of pythagorean word problems worksheet.
