Final answer:
The length of the third side of the triangle is found by subtracting the lengths of the two known sides from the perimeter. The final expression for the length of the third side is -x² - 5xy + 6y - 3y².
Step-by-step explanation:
The question asks for the length of the third side of a triangle given the perimeter and the lengths of the other two sides as algebraic expressions. To find the expression for the third side, we subtract the lengths of the known sides from the perimeter.
The given perimeter is 10x²-3xy+6y.
The lengths of the two known sides are 2x²+2xy and 7x²+3y2.
To find the third side, we equate the sum of the three sides to the perimeter and solve for the third side:
(2x²+2xy) + (7x²+3y2) + third side = 10x² - 3xy + 6y
After combining like terms, the expression for the third side is:
third side = (10x² - 3xy + 6y) - (2x²+2xy) - (7x²+3y2)
By simplifying, we arrive at the final expression for the length of the third side:
third side = 10x² - 3xy + 6y - 2x² - 2xy - 7x² - 3y2
third side = -x² - 5xy + 6y - 3y²