Applying the Pythagorean Theorem Worksheet (teacher made) - Free Printable

Problem Description:

The task involves applying the Pythagorean Theorem to find the length of the hypotenuse of a right-angled triangle. The theorem states:

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

where:
- \( a \) and \( b \) are the lengths of the two legs (shorter sides) of the right-angled triangle.
- \( c \) is the length of the hypotenuse (the longest side, opposite the right angle).

The given values for the legs of the triangle are:
- \( a = 3 \)
- \( b = 4 \)

We need to calculate the length of the hypotenuse \( c \).

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Step-by-Step Solution:

1. Write down the Pythagorean Theorem:
\[
a^2 + b^2 = c^2
\]

2. Substitute the given values \( a = 3 \) and \( b = 4 \):
\[
3^2 + 4^2 = c^2
\]

3. Calculate the squares of \( a \) and \( b \):
\[
3^2 = 9 \quad \text{and} \quad 4^2 = 16
\]

4. Add the squares together:
\[
9 + 16 = 25
\]

5. Set the sum equal to \( c^2 \):
\[
c^2 = 25
\]

6. Solve for \( c \) by taking the square root of both sides:
\[
c = \sqrt{25}
\]

7. Calculate the square root:
\[
c = 5
\]

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Final Answer:

The length of the hypotenuse \( c \) is:
\[
\boxed{5}
\]

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Explanation:

The Pythagorean Theorem is a fundamental relationship in Euclidean geometry that connects the sides of a right-angled triangle. By squaring the lengths of the two legs (\( a \) and \( b \)), adding them together, and then taking the square root of the result, we obtain the length of the hypotenuse (\( c \)). In this case, the calculations confirm that when the legs are 3 and 4, the hypotenuse is 5. This is a classic example of a Pythagorean triple (3, 4, 5).

The diagram provided in the image visually represents the squares of the sides, which can help verify the theorem geometrically. Each square's area corresponds to the square of the side length, and the sum of the areas of the squares on the legs equals the area of the square on the hypotenuse.