Let's solve this problem step-by-step:
1. Define Variables:
- Let [tex]\( s \)[/tex] represent the side length of the square.
2. Area of the Square:
- The area of the square is given by:
[tex]\[
\text{Area of Square} = s^2
\][/tex]
3. Dimensions of the Rectangle:
- The length of the rectangle is 5 units more than the side of the square, which can be expressed as [tex]\( s + 5 \)[/tex].
- The width of the rectangle is half the side of the square, which can be expressed as [tex]\( \frac{s}{2} \)[/tex].
4. Area of the Rectangle:
- The area of the rectangle is given by:
[tex]\[
\text{Area of Rectangle} = \text{Length} \times \text{Width} = (s + 5) \times \frac{s}{2}
\][/tex]
5. Set Up the Equation:
- We are given that the area of the square is equal to the area of the rectangle. Therefore, we equate the two areas:
[tex]\[
s^2 = (s + 5) \times \frac{s}{2}
\][/tex]
6. Simplify the Equation:
- Simplifying the right side of the equation:
[tex]\[
s^2 = s \times \frac{s + 5}{2} = \frac{s(s + 5)}{2}
\][/tex]
- Multiply both sides by 2 to clear the fraction:
[tex]\[
2s^2 = s(s + 5)
\][/tex]
7. Expand and Solve the Equation:
- Expand the right side:
[tex]\[
2s^2 = s^2 + 5s
\][/tex]
- Bring all terms to one side of the equation to set it to zero:
[tex]\[
2s^2 - s^2 - 5s = 0 \implies s^2 - 5s = 0
\][/tex]
8. Factor the Equation:
- Factor the quadratic equation:
[tex]\[
s(s - 5) = 0
\][/tex]
9. Solve for [tex]\( s \)[/tex]:
- Set each factor to zero:
[tex]\[
s = 0 \quad \text{or} \quad s - 5 = 0 \implies s = 5
\][/tex]
- Therefore, the solutions are [tex]\( s = 0 \)[/tex] and [tex]\( s = 5 \)[/tex].
10. Interpret the Solutions:
- The side length [tex]\( s = 0 \)[/tex] is not meaningful in this context because it would imply no square.
- The meaningful solution is [tex]\( s = 5 \)[/tex].
Therefore, the side length of the square is 5 units. The dimensions of the square are [tex]\( 5 \times 5 \)[/tex].
