Pythagorean Theorem Worksheets - Free Printable

It seems like you've uploaded an image containing a worksheet related to the Pythagorean Theorem. Since the image itself isn't directly visible in this context, I'll provide a general explanation of how to solve problems involving the Pythagorean Theorem and then outline the steps for solving typical problems of this type.

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Pythagorean Theorem Overview

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).

This theorem can be used to:
1. Find the length of one side if the other two sides are known.
2. Verify whether a triangle is a right triangle by checking if the equation holds true.

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General Steps to Solve Problems Using the Pythagorean Theorem

1. Identify the given information: Determine which sides are given and which side needs to be found.
2. Assign variables: Label the sides as \(a\), \(b\), and \(c\). Remember that \(c\) is always the hypotenuse.
3. Substitute into the formula: Plug the known values into the equation \(a^2 + b^2 = c^2\).
4. Solve for the unknown: Rearrange the equation to isolate the variable you need to find, and solve using algebra.
5. Check your answer: Ensure that the solution makes sense in the context of the problem (e.g., all side lengths should be positive).

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Example Problem and Solution

Let's assume one of the problems on your worksheet is:

#### Problem Statement:
A right triangle has one leg measuring 6 cm and the other leg measuring 8 cm. Find the length of the hypotenuse.

#### Solution:
1. Identify the given information:
- One leg (\(a\)) = 6 cm
- Other leg (\(b\)) = 8 cm
- Hypotenuse (\(c\)) = ? (to be found)

2. Assign variables:
- \(a = 6\)
- \(b = 8\)
- \(c\) = hypotenuse

3. Substitute into the formula:
\[
a^2 + b^2 = c^2
\]
\[
6^2 + 8^2 = c^2
\]

4. Solve for \(c\):
\[
36 + 64 = c^2
\]
\[
100 = c^2
\]
\[
c = \sqrt{100}
\]
\[
c = 10
\]

5. Check your answer:
- The hypotenuse is 10 cm, which is greater than both legs (6 cm and 8 cm), so the solution is reasonable.

#### Final Answer:
\[
\boxed{10}
\]

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Common Types of Problems

Here are some common types of problems you might encounter on your worksheet, along with how to approach them:

1. Finding the hypotenuse:
- Given the lengths of the two legs (\(a\) and \(b\)), use \(c = \sqrt{a^2 + b^2}\).

2. Finding a leg:
- If the hypotenuse (\(c\)) and one leg (\(a\)) are given, use \(b = \sqrt{c^2 - a^2}\) (or vice versa).

3. Word problems:
- Translate the problem into a right triangle scenario. Identify the legs and hypotenuse, then apply the theorem.

4. Verification problems:
- Check if a triangle with given side lengths is a right triangle by verifying if \(a^2 + b^2 = c^2\).

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If you can provide more specific details or a particular problem from your worksheet, I can tailor the explanation further! For now, the general approach outlined above should help you solve most Pythagorean Theorem problems.

If you have any questions or need clarification, feel free to ask!