Question

1. mathematical connections the perimeters of the two figures are equal.

3x - 4, 2x, 2x, 3x - 4 (for the rhombus); 2x + 6, 2x + 6, ? (for the triangle)
what expression represents the missing side length?

Explanation:

Step1: Calculate perimeter of rhombus

A rhombus has four sides, with two sides of length \(3x - 4\) and two sides of length \(2x\). So its perimeter \(P_{rhombus}=2(3x - 4)+2(2x)\).
Simplify: \(P_{rhombus}=6x - 8 + 4x=10x - 8\).

Step2: Calculate perimeter of triangle

The triangle has two sides of length \(2x + 6\) and one unknown side, let's call it \(y\). So its perimeter \(P_{triangle}=2(2x + 6)+y\).
Simplify: \(P_{triangle}=4x + 12 + y\).

Step3: Set perimeters equal and solve for y

Since \(P_{rhombus}=P_{triangle}\), we have \(10x - 8 = 4x + 12 + y\).
Subtract \(4x + 12\) from both sides: \(y = 10x - 8 - 4x - 12\).
Simplify: \(y = 6x - 20\).

Answer:

The expression for the missing side length is \(6x - 20\).