To find the length of the third side of the triangle, follow these steps:
1. Identify the Perimeter and the Given Sides:
The perimeter of the triangle is given as [tex]\( 7x + 8 \)[/tex].
The first side of the triangle has a length of [tex]\( 3x \)[/tex].
The second side of the triangle has a length of [tex]\( 2x - 5 \)[/tex].
2. Set Up the Perimeter Equation:
We know that the perimeter of a triangle is the sum of the lengths of its three sides. Let's call the third side [tex]\( \text{side3} \)[/tex]. Therefore, we have:
[tex]\[
\text{side1} + \text{side2} + \text{side3} = \text{perimeter}
\][/tex]
Plugging in the known values, we get:
[tex]\[
3x + (2x - 5) + \text{side3} = 7x + 8
\][/tex]
3. Combine and Simplify the Known Terms:
Let's simplify the left-hand side by combining like terms:
[tex]\[
3x + 2x - 5 + \text{side3} = 7x + 8
\][/tex]
This simplifies to:
[tex]\[
5x - 5 + \text{side3} = 7x + 8
\][/tex]
4. Isolate the Third Side:
To find [tex]\( \text{side3} \)[/tex], isolate it on one side of the equation. Subtract [tex]\( 5x - 5 \)[/tex] from both sides:
[tex]\[
\text{side3} = 7x + 8 - (5x - 5)
\][/tex]
Simplify the right-hand side:
[tex]\[
\text{side3} = 7x + 8 - 5x + 5 = 2x + 13
\][/tex]
Therefore, the length of the third side of the triangle is [tex]\( 2x + 13 \)[/tex].
5. Verify the Answer with the Given Options:
From the given options:
[tex]\[
x + 13, \quad 2x + 13, \quad 5x - 5, \quad 12x + 3
\][/tex]
we find that [tex]\( 2x + 13 \)[/tex] matches our calculated length for the third side.
Thus, the correct answer is [tex]\( 2x + 13 \)[/tex].
So, the length of the third side is [tex]\( 2x + 13 \)[/tex] and this matches option 2.
