We have a geometric problem about finding the optimal location for a library. Two schools are located at points C and D, with perpendicular distances to line AB. We need to find point E on line AB such that the distances from E to both schools are equal.
Now let's set up a coordinate system to solve this problem algebraically. We place point A at the origin, so A is at coordinates zero comma zero. Point B is 2.5 kilometers to the right at coordinates 2.5 comma zero. Point C is 1.5 kilometers below A at coordinates zero comma negative 1.5. Point D is 1 kilometer below B at coordinates 2.5 comma negative 1. The library location E is at coordinates x comma zero, where x is what we need to find.
To solve this problem, we need to apply the distance formula. The distance from C to E is the square root of x minus zero squared plus zero minus negative 1.5 squared, which simplifies to the square root of x squared plus 2.25. The distance from D to E is the square root of x minus 2.5 squared plus zero minus negative 1 squared, which simplifies to the square root of x minus 2.5 squared plus 1. For the library to be equidistant from both schools, these two distances must be equal.
Now let's solve the equation step by step. We start with CE equals DE. Substituting our distance formulas, we get square root of x squared plus 2.25 equals square root of x minus 2.5 squared plus 1. Squaring both sides eliminates the square roots. Expanding the right side gives us x squared minus 5x plus 6.25 plus 1. Simplifying, we get 2.25 equals negative 5x plus 7.25. Solving for x, we find that 5x equals 5, so x equals 1. Therefore, the library should be built 1 kilometer from point A.
Let's verify our solution. With E at position 1 comma 0, the distance from A to E is exactly 1 kilometer. The distance from C to E is the square root of 1 squared plus 1.5 squared, which equals the square root of 3.25. The distance from D to E is the square root of 1.5 squared plus 1 squared, which also equals the square root of 3.25. Since both distances are equal at approximately 1.80 kilometers, our solution is correct. Therefore, the library should be built 1 kilometer from point A to ensure it is equidistant from both schools.