Question

the two triangles shown have the same perimeters. what is the perimeter of each triangle?

Explanation:

Step1: Recall perimeter formula

The perimeter \(P\) of a triangle is the sum of the lengths of its sides. For the first triangle with side - lengths \(4x\), \(2x + 1\), and \(6x+3\), the perimeter \(P_1\) is \(P_1=4x+(2x + 1)+(6x + 3)\).

Step2: Combine like - terms for first triangle

\[

$$\begin{align*} P_1&=(4x+2x + 6x)+(1 + 3)\\ &=12x+4 \end{align*}$$

\]

Step3: Calculate perimeter of second triangle

For the second triangle with side - lengths \(3x + 2\), \(5x\), and \(5x\), the perimeter \(P_2\) is \(P_2=(3x + 2)+5x+5x\).

Step4: Combine like - terms for second triangle

\[

$$\begin{align*} P_2&=(3x+5x + 5x)+2\\ &=13x+2 \end{align*}$$

\]

Step5: Set perimeters equal

Since the two triangles have the same perimeters, \(P_1 = P_2\), so \(12x+4=13x + 2\).

Step6: Solve for \(x\)

Subtract \(12x\) from both sides: \(4=x + 2\). Then subtract 2 from both sides to get \(x = 2\).

Step7: Find the perimeter

Substitute \(x = 2\) into the formula for \(P_1\) (we could also use \(P_2\)). \(P_1=12x+4\), so \(P_1=12\times2+4=24 + 4=28\).

Answer:

28