Question

a group of students is given a 10 by 10 grid to cut into individual unit squares. the challenge is to create two squares using all of the unit squares. their teacher states that after the two new squares are formed, one should have a side length two units greater than the other. which equation represents x, the side length of the greater square? \\(\bigcirc\\) \\(x^2 + (x - 2)^2 = 10\\) \\(\bigcirc\\) \\(x^2 + 2x^2 = 10\\) \\(\bigcirc\\) \\(x^2 + (x - 2)^2 = 100\\) \\(\bigcirc\\) \\(x^2 + 2x^2 = 100\\)

Explanation:

Step1: Determine total unit squares

A 10 by 10 grid has \(10\times10 = 100\) unit squares. So total area of two squares is 100.

Step2: Define side lengths of squares

Let \(x\) be side length of greater square. Then smaller square has side length \(x - 2\) (since greater is 2 units more).

Step3: Area of each square

Area of greater square: \(x^{2}\), area of smaller square: \((x - 2)^{2}\).

Step4: Set up equation

Sum of areas equals total unit squares: \(x^{2}+(x - 2)^{2}=100\).

Answer:

\(x^{2}+(x - 2)^{2}=100\) (the third option)