Question

In this proposition, demonstrate the mathematical proof that four times the rectangle formed by AB and BC, added to the square on AC, is equal to the square described on AB. Provide a step-by-step proof for this geometric statement."

Answer 1

A square with side lengths twice that of another square has an area four times greater. The area comparison follows the square of the scale factor rule, which in this case leads to a ratio of areas of 1:4 between the smaller and larger square, respectively.

Step-by-step explanation:

The question is asking about the relationship between the area of two squares, one with sides twice as long as the other. If we have a square with a side length of 4 inches, the area of this square is 4 inches × 4 inches, which equals 16 square inches. If another square has dimensions twice that of the first, its side length is 4 inches × 2, totaling 8 inches. The area of the larger square is then 8 inches × 8 inches, which equals 64 square inches.

Since the side lengths of the squares are in the ratio of 1:2, the areas will be in the ratio of the squares of these numbers, which is 1:4. Therefore, the area of the larger square is 4 times larger than the area of the smaller square. This demonstrates the scale factor in action when comparing the areas of similar shapes.