Write the sides of ∆ABC in order from LONGEST to SHORTEST if m∠A = x² - 18, m∠B = 2x² + 3x, and m∠C = 12x.

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  b, c, a   or   AC, AB, BC

Step-by-step explanation:

The sum of the angle measures is 180°, so we have ...

  ∠A +∠B +∠C = 180°

  (x^2 -18) +(2x^2 +3x) +(12x) = 180

  3x^2 +15x = 198 . . . . add 18, simplify

  x^2 +5x = 66 . . . . . . divide by 3

  x^2 +5x -66 = 0 . . . subtract 66 to put in standard form

  (x +11)(x -6) = 0 . . . . . factor

Solutions are x = -11, x = 6. Angle C requires x > 0, so the only useful solution is x = 6.

Then the angle measures are ...

  ∠A = x^2 -18 = 6^2 -18 = 18

  ∠B = 2x^2 +3x = x(2x +3) = 6(2·6 +3) = 90

  ∠C = 12x = 12·6 = 72

The angles, largest to smallest are B, C, A. Their opposite sides are in the same order, longest to shortest: b, c, a. Those sides will also be named AC, AB, BC.

step-by-step explanation:

most likely the answer is a.

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