I need to know the Pythagorean theorem and word problems.
Understand the Problem
The question is asking for an explanation of the Pythagorean theorem along with examples of word problems that illustrate its application. This entails defining the theorem and demonstrating its use in real-world scenarios.
Answer
The length of the ladder is 5 feet.
Answer for screen readers
The length of the ladder is 5 feet.
Steps to Solve
- 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:
$$ c^2 = a^2 + b^2 $$
- 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.
- 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.
- 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:
$$ c^2 = a^2 + b^2 $$ $$ c^2 = 3^2 + 4^2 $$ $$ c^2 = 9 + 16 $$ $$ c^2 = 25 $$
- 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.
The length of the ladder is 5 feet.
More Information
The Pythagorean theorem is widely used in various fields, such as architecture, navigation, and even in computer graphics, where right triangles often appear in models.
Tips
- Forgetting to identify which side is the hypotenuse can lead to incorrect calculations. Remember, the hypotenuse is always opposite the right angle.
- Confusing the sides; always check that you are using the correct measurements for $a$ and $b$.