A square and a triangle have equal perimeters. The square has a perimeter of $4(2 p-4)$, and the triangle has a perimeter of $4 p+36$. Find the perimeter value.
Answer (1)
Set the perimeter of the square equal to the perimeter of the triangle: 4 ( 2 p − 4 ) = 4 p + 36 .
Solve the equation for p : p = 13 .
Substitute the value of p into either expression for the perimeter to find the perimeter value.
The perimeter value is 88 .
Explanation
Understanding the Problem We are given that a square and a triangle have equal perimeters. The perimeter of the square is given by 4 ( 2 p − 4 ) , and the perimeter of the triangle is given by 4 p + 36 . We need to find the value of the perimeter.
Setting up the Equation Since the perimeters are equal, we can set up the equation: 4 ( 2 p − 4 ) = 4 p + 36
Distributing Now, we solve for p . First, distribute the 4 on the left side: 8 p − 16 = 4 p + 36
Subtracting 4p Subtract 4 p from both sides: 4 p − 16 = 36
Adding 16 Add 16 to both sides: 4 p = 52
Dividing by 4 Divide both sides by 4 : p = 13
Finding the Perimeter Now that we have the value of p , we can find the perimeter by substituting p = 13 into either expression. Let's use the expression for the perimeter of the triangle: P er im e t er = 4 p + 36 = 4 ( 13 ) + 36 = 52 + 36 = 88
Final Answer Therefore, the perimeter value is 88 .
Examples
Imagine you're designing a garden with a square flower bed and a triangular herb patch. You want both to have the same border length to save on edging material. By setting their perimeter formulas equal, you can determine the exact border length needed for both, ensuring a balanced and aesthetically pleasing garden design. This problem demonstrates how algebraic equations help in practical design and resource management.
