Final answer:
The student's question relates to the Pythagorean theorem and comparing areas of squares on the sides of a right triangle. The area of the largest square formed on the hypotenuse is equal to the sum of the areas of the two smaller squares on the other sides. An example with side lengths shows that the ratio of the areas is the square of the scale factor.
Step-by-step explanation:
The question involves understanding the Pythagorean theorem which relates the sides of a right triangle and helps in comparing the areas of squares constructed on these sides.
When three squares are joined at their vertices to form a right triangle, the area of the largest square is equal to the sum of the areas of the two smaller squares.
This is represented by the relationship: a² + b² = c², where 'a' and 'b' are the lengths of the legs of the triangle and 'c' is the length of the hypotenuse.
To understand how the area of a larger square compares to that of a smaller square, consider an example where side 'a' is 4 inches and the larger square has sides twice as long (8 inches).
The ratio of their areas will be the square of the scale factor, which in this case is 4, because (8/4)² = 4.
Therefore, the area of the larger square is 4 times the area of the smaller square.