Question

8. m∠A=9x−7,m∠B=7x−9, and m∠C=28−2x. List the sides of △ABC in order from shorient to longest.

Answer 1

To list the sides of △ABC in order from shortest to longest, use the Triangle Inequality Theorem and solve for x to find the angles. Then, use the angles to determine the lengths of the sides. The order from shortest to longest is AB, BC, AC.

Step-by-step explanation:

To list the sides of △ABC in order from shortest to longest, we need to use the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Using the given information, we can set up three inequalities and solve for x to find the values of the angles. Then, we can use the values of the angles to determine the lengths of the sides.

1. Start by setting up the inequalities: 9x - 7 + 7x - 9 > 28 - 2x, 9x - 7 + 28 - 2x > 7x - 9, and 7x - 9 + 28 - 2x > 9x - 7.
2. Simplify the inequalities: 16x - 16 > 28 - 2x, 16x + 19 > 7x - 7, and 5x + 19 > 9x - 7.
3. Solve the inequalities: x > 2, x > -13, and x > 6.
4. Since x > 6, we can substitute this value into the given equations to find the measures of the angles: m∠A = 9(6) - 7 = 47°, m∠B = 7(6) - 9 = 33°, and m∠C = 28 - 2(6) = 16°.
5. The longest side of a triangle corresponds to the largest angle, so the order from shortest to longest is AB, BC, AC.

Learn more about Triangle Inequality Theorem