Solving the Ladder and House Problem Using Pythagoras Theorem

When dealing with a ladder leaning against a house, a common question is to find out how far the base of the ladder should be from the side of the house for optimal safety and stability. This article will explore how to solve such a problem using the Pythagorean Theorem and simple geometry principles.

Understanding the Problem

The problem at hand involves a 50-foot ladder leaning against a house, with the top of the ladder 37 feet up the side of the house. The goal is to determine the distance from the base of the ladder to the side of the house. This problem can be solved by recognizing that the situation forms a right triangle where the ladder acts as the hypotenuse, the height of the house forms one of the legs, and the distance from the base of the ladder to the wall is the other leg.

Applying the Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is expressed as:

a^2 b^2 c^2

In our scenario, let:

c 50 feet (the length of the ladder, or the hypotenuse) a 37 feet (the height the ladder reaches up the side of the house) b the distance from the base of the ladder to the side of the house (the unknown leg we need to find)

Substituting these values into the Pythagorean Theorem:

37^2 b^2 50^2

Calculating the squares:

1369 b^2 2500

Isolating b^2:

b^2 2500 - 1369 1131

Finally, taking the square root of both sides to solve for b:

b sqrt{1131} 33.63 text{ feet}

This means the base of the ladder is approximately 33.63 feet from the side of the house.

Practical Application

Understanding the distance the ladder should be from the house is crucial for safety and stability. According to certain guidelines, a ladder's base should typically be set away from the house at a distance of about one-third to one-fourth the length of the ladder for optimal stability. In this case, for a 50-foot ladder, this would mean setting the base about 12.5 feet away from the house.

However, in specific instances, the actual setup distance might vary. For instance, if the ladder is 50 feet high but only extends 37 feet up the side of the house, the base should be approximately 33.63 feet from the house. The standard advice of 12 feet applies to a ladders typical setup, where the base should be a quarter of the ladder's reach (50 ft / 4 12.5 ft), but in this context, the precise distance is clearly 33.63 feet.

Using the Pythagorean Theorem, we can see that while the base of a 50-foot ladder might need to be set 12 feet away for a 37-foot height setup, the actual required distance is significantly greater, at approximately 33.63 feet.

Conclusion

To summarize, the distance from the base of a 50-foot ladder to the side of a house, if the ladder reaches 37 feet up the house, is approximately 33.63 feet. This is important for both practical setup and adhering to safety guidelines. The use of the Pythagorean Theorem provides a precise and reliable method for solving such problems efficiently.