Final answer:
To list the sides of triangle ABC from shortest to longest, we calculate each angle using the given angle measurements and then apply the triangle inequality theorem. This theorem states that the smallest angle is opposite the shortest side, and the largest angle is opposite the longest side. The order from shortest to longest side is AC, AB, and BC.
Step-by-step explanation:
To determine the lengths of the sides of triangle ABC in order from shortest to longest, we need to first solve for the values of x in mA, mB, and mC, which represent the measures of the angles at each vertex. Given mA = 8x – 2, mB = 2x – 8, and mC = 94 – 4x, we can take into account that the sum of the angles in any triangle is 180 degrees. Thus, we can write the equation:
8x – 2 + 2x – 8 + 94 – 4x = 180
Simplifying, we get:
6x + 84 = 180
Subtracting 84 from both sides gives us:
6x = 96
Dividing by 6, we find:
x = 16
Using the value of x, we calculate the measure of each angle:
mA = 8(16) – 2 = 126
mB = 2(16) – 8 = 24
mC = 94 – 4(16) = 30
According to the triangle inequality theorem, the smallest angle is opposite the shortest side, and the largest angle is opposite the longest side. Therefore, side AC is opposite angle B, side AB is opposite angle C, and side BC is opposite angle A.
So the order from shortest to longest is:
Side AC
Side AB
Side BC