I need to know the Pythagorean theorem and word problems.

Steps to Solve

1. Define the Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse ($c$) is equal to the sum of the squares of the lengths of the other two sides ($a$ and $b$). This can be expressed mathematically as:

$$ c^2 = a^2 + b^2 $$

2. Explain each component of the theorem

In the theorem:

• $c$ is the length of the hypotenuse (the side opposite the right angle).
• $a$ and $b$ are the lengths of the other two sides of the triangle.

3. Example Word Problem

Let's consider a word problem: A ladder is leaning against a wall. The base of the ladder is 4 feet away from the wall, and the top of the ladder reaches a height of 3 feet on the wall. To find the length of the ladder ($c$), we use the Pythagorean theorem.

4. Set up the equation for the word problem

Here, we identify:

• $a = 3$ feet (height of the wall)
• $b = 4$ feet (distance from the wall)

Using the Pythagorean theorem:

$$ c^2 = a^2 + b^2 $$ $$ c^2 = 3^2 + 4^2 $$ $$ c^2 = 9 + 16 $$ $$ c^2 = 25 $$

5. Calculate the length of the ladder

Now we solve for $c$:

$$ c = \sqrt{25} $$ $$ c = 5 $$

Thus, the length of the ladder is 5 feet.