Discover the True Length of Your Ladder Using Pythagorean Theorem

The Pythagorean Theorem is one of the most fundamental principles in mathematics, and is particularly useful in many practical applications. In this article, we will explore how to use the Pythagorean Theorem to solve problems involving feet, ladders, and walls.

What is the Pythagorean Theorem?

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In other words:

c² = a² + b²

where c is the length of the hypotenuse, and a and b are the lengths of the other two sides (the legs).

Feet and Ladder Problems

One of the most common applications of the Pythagorean Theorem is in solving problems involving ladders or other inclined objects. For example, suppose you have a ladder leaning against a wall. You want to know how far up the wall the ladder will reach if the bottom of the ladder is a feet from the wall and the ladder itself is b feet long. Using the Pythagorean Theorem, we can solve for the height:

c² = a² + b²

c = √(a² + b²)

For example, if the ladder is 20 feet long and the bottom of the ladder is 16 feet from the wall, we can calculate the height to be:

c = √(16² + 20²) = √(256 + 400) = √656 ≈ 25.6 feet

So the ladder will reach a height of approximately 25.6 feet up the wall.

Wall Problems

The Pythagorean Theorem is also useful in solving problems involving walls. For example, suppose you want to know how far apart two points on a wall are, but you can't measure the distance directly. Instead, you measure the distances from each point to a third point on the wall, and you know that the three points form a right-angled triangle. Using the Pythagorean Theorem, we can solve for the distance between the two points:

c² = a² + b²

c = √(a² + b²)

For example, suppose that point A is 5 feet from the third point on the wall, and point B is 8 feet from the third point on the wall. You can calculate the distance between A and B as:

c = √(5² + 8²) = √(25 + 64) = √89 ≈ 9.4 feet

So the distance between points A and B on the wall is approximately 9.4 feet.

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