The longest side of triangle ABC is opposite angle C, and the corresponding expression is 13x - 17.
To determine the longest side of triangle ABC, we can use the Law of Cosines. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides a, b, and c, and an angle C opposite side c, the Law of Cosines is given by:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we need to find the longest side, so we are interested in the side opposite the largest angle. Let's find the angle measures first:
m∠A = x^2 + 12
m∠B = 11x + 5
m∠C = 13x - 17
Now, to find the longest side, we need to find the maximum value among these angle measures. Taking derivatives of each expression with respect to x and setting them equal to zero can help find critical points. However, this may lead to complex solutions.
Alternatively, we can compare the coefficients of x in each expression. The angle with the largest coefficient of x tends to dominate as x becomes large. Therefore, the longest side is likely opposite the angle with the largest coefficient.
Comparing the coefficients:
x^2 + 12 (Coefficient of x^2: 1)
11x + 5 (Coefficient of x: 11)
13x - 17 (Coefficient of x: 13)
Therefore, m∠C = 13x - 17 has the largest coefficient of x, and the longest side is likely opposite angle C.