Pythagorean Theorem worksheet with problems and diagrams for solving right triangle side lengths.

Problem Analysis:

The problem involves applying the Pythagorean Theorem to a right triangle. The Pythagorean Theorem states that for a right triangle with legs of lengths \( a \) and \( b \), and hypotenuse of length \( c \):

\[
a^2 + b^2 = c^2
\]

We are given a right triangle \( \triangle ABC \) with sides labeled as follows:
- One leg is \( a \).
- The other leg is \( b \).
- The hypotenuse is \( c \).

We need to solve for missing values using the Pythagorean Theorem.

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Step-by-Step Solution:

#### Part 1: Solve for \( c \) when \( a = 3 \) and \( b = 4 \)

1. Given: \( a = 3 \), \( b = 4 \)
2. Formula: \( a^2 + b^2 = c^2 \)
3. Substitute the values:
\[
3^2 + 4^2 = c^2
\]
4. Simplify:
\[
9 + 16 = c^2
\]
\[
25 = c^2
\]
5. Solve for \( c \):
\[
c = \sqrt{25} = 5
\]

Answer: \( c = 5 \)

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#### Part 2: Solve for \( b \) when \( a = 6 \) and \( c = 10 \)

1. Given: \( a = 6 \), \( c = 10 \)
2. Formula: \( a^2 + b^2 = c^2 \)
3. Substitute the values:
\[
6^2 + b^2 = 10^2
\]
4. Simplify:
\[
36 + b^2 = 100
\]
5. Isolate \( b^2 \):
\[
b^2 = 100 - 36
\]
\[
b^2 = 64
\]
6. Solve for \( b \):
\[
b = \sqrt{64} = 8
\]

Answer: \( b = 8 \)

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#### Part 3: Solve for \( a \) when \( b = 12 \) and \( c = 13 \)

1. Given: \( b = 12 \), \( c = 13 \)
2. Formula: \( a^2 + b^2 = c^2 \)
3. Substitute the values:
\[
a^2 + 12^2 = 13^2
\]
4. Simplify:
\[
a^2 + 144 = 169
\]
5. Isolate \( a^2 \):
\[
a^2 = 169 - 144
\]
\[
a^2 = 25
\]
6. Solve for \( a \):
\[
a = \sqrt{25} = 5
\]

Answer: \( a = 5 \)

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Final Answers:

1. When \( a = 3 \) and \( b = 4 \), \( c = \boxed{5} \).
2. When \( a = 6 \) and \( c = 10 \), \( b = \boxed{8} \).
3. When \( b = 12 \) and \( c = 13 \), \( a = \boxed{5} \).