Pythagorean Theorem Worksheets - Free Printable

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The Pythagorean Theorem states:
$$
a^2 + b^2 = c^2
$$
where:
- \( a \) and \( b \) are the lengths of the two legs of a right triangle,
- \( c \) is the length of the hypotenuse (the side opposite the right angle).

General Steps to Solve Problems Involving the Pythagorean Theorem:

1. Identify the given information: Which sides are known, and which side needs to be found.
2. Determine whether you need to find the hypotenuse (\( c \)) or one of the legs (\( a \) or \( b \)).
3. Substitute the known values into the formula \( a^2 + b^2 = c^2 \).
4. Solve for the unknown side using algebraic manipulation.

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Example Problem Types and Solutions:

#### Problem Type 1: Finding the Hypotenuse
If the lengths of the two legs of a right triangle are given, you can find the hypotenuse.

Example:
A right triangle has legs of lengths 3 units and 4 units. Find the length of the hypotenuse.

Solution:
1. Identify the given values: \( a = 3 \), \( b = 4 \).
2. Use the Pythagorean Theorem: \( a^2 + b^2 = c^2 \).
$$
3^2 + 4^2 = c^2
$$
3. Simplify:
$$
9 + 16 = c^2
$$
$$
25 = c^2
$$
4. Solve for \( c \):
$$
c = \sqrt{25} = 5
$$

Answer: The hypotenuse is \( \boxed{5} \).

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#### Problem Type 2: Finding a Leg
If the hypotenuse and one leg are given, you can find the other leg.

Example:
A right triangle has a hypotenuse of 10 units and one leg of 6 units. Find the length of the other leg.

Solution:
1. Identify the given values: \( c = 10 \), \( a = 6 \).
2. Use the Pythagorean Theorem: \( a^2 + b^2 = c^2 \).
$$
6^2 + b^2 = 10^2
$$
3. Simplify:
$$
36 + b^2 = 100
$$
4. Isolate \( b^2 \):
$$
b^2 = 100 - 36
$$
$$
b^2 = 64
$$
5. Solve for \( b \):
$$
b = \sqrt{64} = 8
$$

Answer: The other leg is \( \boxed{8} \).

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#### Problem Type 3: Real-World Application
Problems may involve real-world scenarios where the Pythagorean Theorem can be applied.

Example:
A ladder leans against a wall, reaching a height of 12 feet. The base of the ladder is 5 feet away from the wall. How long is the ladder?

Solution:
1. Identify the given values: The height of the wall (\( a \)) is 12 feet, and the distance from the wall to the base of the ladder (\( b \)) is 5 feet. The length of the ladder is the hypotenuse (\( c \)).
2. Use the Pythagorean Theorem: \( a^2 + b^2 = c^2 \).
$$
12^2 + 5^2 = c^2
$$
3. Simplify:
$$
144 + 25 = c^2
$$
$$
169 = c^2
$$
4. Solve for \( c \):
$$
c = \sqrt{169} = 13
$$

Answer: The length of the ladder is \( \boxed{13} \) feet.

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Additional Notes:

- Always check that the triangle is a right triangle before applying the Pythagorean Theorem.
- Ensure that the units are consistent throughout the problem.
- If the problem involves fractions or decimals, simplify carefully.

If you can provide more specific details about the problems in your image, I can tailor the solutions accordingly! Otherwise, the above examples should help you tackle similar problems.

Let me know if you need further clarification!