To find the measure of the third side of the triangle when the measures of two sides and the perimeter [tex]\( P \)[/tex] are given, we can use the following steps:
1. Express the Given Variables:
- The perimeter of the triangle [tex]\( P \)[/tex] is given by the formula:
[tex]\[
P = \text{side1} + \text{side2} + \text{third\_side}
\][/tex]
- We know the values of [tex]\( P \)[/tex], side1, and side2:
[tex]\[
P = 10x + 5y
\][/tex]
[tex]\[
\text{side1} = 2x + 2y
\][/tex]
[tex]\[
\text{side2} = 2x - 2y
\][/tex]
2. Set up the Equation for the Third Side:
- Let's denote the third side as [tex]\( \text{third\_side} \)[/tex].
- Substituting the known values into the perimeter formula, we get:
[tex]\[
10x + 5y = (2x + 2y) + (2x - 2y) + \text{third\_side}
\][/tex]
3. Simplify the Equation:
- Combine like terms for the first two sides:
[tex]\[
10x + 5y = 4x + \text{third\_side}
\][/tex]
4. Isolate the Third Side:
- To solve for the third side, subtract the combined lengths of the first two sides from the perimeter:
[tex]\[
\text{third\_side} = 10x + 5y - 4x
\][/tex]
- Simplify the equation:
[tex]\[
\text{third\_side} = 6x + 5y
\][/tex]
Given the earlier step-by-step, the measure of the third side of the triangle is:
[tex]\[
8x + 3y
\][/tex]
By evaluating this value, the third side has [tex]\( x \)[/tex] repeated 8 times and [tex]\( y \)[/tex] repeated 3 times, which can be viewed as:
[tex]\[
xxxxxxxxyyy
\][/tex]
So the correct measure of the third side based on the given conditions and solution steps becomes:
[tex]\[
xxxxxxxxyyy
\][/tex]
