One square has twice the area of another square. Explain why it is impossible for both squares to have side lengths that are whole numbers.
The areas of two similar squares are proportional to the square of their corresponding sides
let the side of the smaller be x and the side of the larger be y
then x^2 : y^2= 1 : 2
2x^2 = y^2
y = (√2)(x)
suppose x is a whole number, then √2 times a whole number cannot be a whole number.
To explain why it's impossible for both squares to have whole number side lengths, let's assume the smaller square has a side length of "x" units. The larger square, which has twice the area, would need a side length of "2x" units.
The area of a square is calculated by multiplying the length of one of its sides by itself. So, the area of the smaller square would be x * x, or x^2 units. The area of the larger square would be (2x) * (2x), which simplifies to 4x^2 units.
Given that the area of the larger square is twice that of the smaller square, we can set up the following equation:
4x^2 = 2 * x^2
If we simplify this equation, we get:
4x^2 = 2x^2
Now, let's subtract 2x^2 from both sides:
2x^2 = 0
Dividing both sides by 2, we find:
x^2 = 0
Here's where the problem arises. We know that the side length of a square represents a positive distance, so the value of x cannot be equal to zero. Therefore, there are no positive whole number solutions for x in this equation.
Hence, it is impossible for both squares to have whole number side lengths.