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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 [tex]\(x\)[/tex], the side length of the greater square?

A. [tex]\(x^2 + (x-2)^2 = 10\)[/tex]

B. [tex]\(x^2 + 2x^2 = 10\)[/tex]

C. [tex]\(x^2 + (x-2)^2 = 100\)[/tex]

D. [tex]\(x^2 + 2x^2 = 100\)[/tex]

Sagot :

GinnyAnsweridn
To solve this problem, let's first understand the conditions and variables involved:

1. Grid Size and Total Area: The given grid has dimensions 10 by 10, which means it consists of a total of [tex]\(10 \times 10 = 100\)[/tex] unit squares.

2. New Squares Side Lengths: We need to create two new squares using these 100 unit squares such that the side length of one square is 2 units greater than the side length of the other.

Let's define:
- [tex]\( x \)[/tex] as the side length of the greater square.
- [tex]\( x - 2 \)[/tex] as the side length of the smaller square (since it is 2 units less than [tex]\( x \)[/tex]).

Next, we need to express the total area of these squares using their side lengths:
- The area of the greater square will be [tex]\( x^2 \)[/tex].
- The area of the smaller square will be [tex]\((x - 2)^2\)[/tex].

Since the total area must add up to 100 unit squares, we can set up the following equation:

[tex]\[ x^2 + (x - 2)^2 = 100 \][/tex]

This equation represents the relationship between the side lengths of the two squares and the total area. Let's verify the options provided:

1. [tex]\( x^2 + (x - 2)^2 = 10 \)[/tex]
2. [tex]\( x^2 + 2 x^2 = 10 \)[/tex]
3. [tex]\( x^2 + (x - 2)^2 = 100 \)[/tex]
4. [tex]\( x^2 + 2 x^2 = 100 \)[/tex]

From our setup, we identified that the correct equation must be [tex]\( x^2 + (x - 2)^2 = 100 \)[/tex], which matches the third option.

So, the correct equation that represents [tex]\( x \)[/tex], the side length of the greater square, is:
[tex]\[ x^2 + (x - 2)^2 = 100 \][/tex]
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