To solve the problem, let's first outline the given information and set up the necessary equations:
A square has side length (x). A rectangle has length ((x+1)) and width ((x-1)). An equilateral triangle has side length equal to the square's side. If the perimeter of the triangle is 5 less than the perimeter of the rectangle, find the value of (x).
Perimeter of the rectangle:
The perimeter of a rectangle is (2 \times (\text{length} + \text{width})).
[
P_{\text{rectangle}} = 2[(x+1) + (x-1)] = 2(2x) = 4x
]
Perimeter of the equilateral triangle:
The perimeter of an equilateral triangle is (3 \times \text{side length}).
[
P_{\text{triangle}} = 3x
]
Relate the perimeters:
Given that the triangle's perimeter is 5 less than the rectangle's perimeter:
[
3x = 4x - 5
]
Solve for (x):
Subtract (3x) from both sides:
[
0 = x - 5 \implies x = 5
]
Answer: (\boxed{5})
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