Question

the table shows the side lengths of a triangle.
the perimeter of the triangle is 6a + 3 units. write an expression in simplest form for the length of side 3.

Explanation:

Step1: Recall the formula for the perimeter of a triangle

The perimeter \(P\) of a triangle is \(P = s_1 + s_2 + s_3\), where \(s_1\), \(s_2\), and \(s_3\) are the side - lengths of the triangle. We know that \(P=6a + 3\), \(s_1 = 2(a + 3)\), and \(s_2=3a - 1\).

Step2: Expand \(s_1\)

Expand \(s_1 = 2(a + 3)\) using the distributive property \(a(b + c)=ab+ac\). So, \(s_1=2a+6\).

Step3: Substitute into the perimeter formula

Substitute \(P\), \(s_1\), and \(s_2\) into \(P = s_1 + s_2 + s_3\). We get \(6a + 3=(2a + 6)+(3a - 1)+s_3\).

Step4: Simplify the right - hand side

Combine like terms on the right - hand side: \((2a+3a)+(6 - 1)+s_3=5a + 5+s_3\). So, the equation becomes \(6a + 3=5a + 5+s_3\).

Step5: Solve for \(s_3\)

Subtract \(5a + 5\) from both sides of the equation. \(s_3=(6a + 3)-(5a + 5)\).
Using the distributive property \(a-(b + c)=a - b - c\), we have \(s_3=6a+3 - 5a - 5\).
Combine like terms: \((6a-5a)+(3 - 5)=a - 2\).

Answer:

\(a - 2\)