Length of a Ladder A building is 2 ft from a 9 -ft fence that surrounds the property. A worker w

step 1 This problem we have been given that a building is 2 feet from a 9 feet fence that surrounds the

Length of a Ladder A building is 2 ft from a 9 -ft fence...

Length of a Ladder A building is $2 \mathrm{ft}$ from a 9 -ft fence that surrounds the property. A worker wants to wash a window in the building 13 ft from the ground. He plans to place a ladder over the fence so it rests against the building. (See the figure.) He decides he should place the ladder 8 ft from the fence for stability. To the nearest tenth of a foot, how long a ladder will he need?

Key Concepts

Coordinate Geometry in Practical Problems

Using a coordinate system to model positions such as the base of a ladder, an obstacle (like a fence), and the target point (like a window) allows for a precise geometric interpretation of the problem. By assigning coordinates to these key points, one can compute distances and slopes, set up relationships between them, and ultimately solve for the unknown quantities, such as the ladder length.

Decomposition of a Ladder Problem

Many ladder problems involve obstacles that force the ladder to 'bend' over an intermediate point, even though the ladder itself remains straight. This requires decomposing the problem into segments that form separate right triangles with shared points. By analyzing these individual triangles and how they connect, one can set up equations that relate the different segments to the total ladder length.

Right Triangle Geometry

This concept involves understanding that when a ladder leans against a vertical wall, it forms a right triangle with the wall and the ground. In such triangles, one angle is 90° and the lengths of the sides are related by specific geometric properties, making it a fundamental model for many real?world problems involving heights and distances.

Pythagorean Theorem

The Pythagorean theorem is a key tool in solving right triangle problems. It states that in a right triangle the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem enables calculation of an unknown side length when the other two are known, which is essential in determining the length of a ladder in various configurations.

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