Question

problem #6: below the perimeter of the rectangle is equal to the perimeter of the triangle. using an equation, find the perimeter of both figures. what are you trying to find out: workspace: solution:

Explanation:

Step1: Calculate perimeter of rectangle

The formula for the perimeter of a rectangle is \( P = 2\times(\text{length} + \text{width}) \). For the rectangle with length \( 2x + 3 \) and width \( x - 1 \), the perimeter \( P_{rect} = 2[(2x + 3)+(x - 1)] \). Simplify inside the brackets: \( (2x + 3)+(x - 1) = 3x + 2 \). Then \( P_{rect} = 2(3x + 2)=6x + 4 \).

Step2: Calculate perimeter of triangle

The perimeter of a triangle is the sum of its three sides. For the triangle with sides \( x - 5 \), \( 4x - 3 \), and \( 3x - 2 \), the perimeter \( P_{tri}=(x - 5)+(4x - 3)+(3x - 2) \). Combine like terms: \( x + 4x + 3x - 5 - 3 - 2 = 8x - 10 \).

Step3: Set perimeters equal and solve for x

Since the perimeters are equal, set \( 6x + 4 = 8x - 10 \). Subtract \( 6x \) from both sides: \( 4 = 2x - 10 \). Add 10 to both sides: \( 14 = 2x \). Divide by 2: \( x = 7 \).

Step4: Find perimeter of rectangle (or triangle)

Substitute \( x = 7 \) into \( P_{rect} = 6x + 4 \): \( 6(7)+4 = 42 + 4 = 46 \). (Or into \( P_{tri}=8x - 10 \): \( 8(7)-10 = 56 - 10 = 46 \).)

Answer:

The perimeter of both figures is \(\boldsymbol{46}\).