Multiple choice

The two shorter sides of a triangle measure $9$ and $18$. If the internal angle bisector drawn to the longest side measures $8$, find the measure of the longest side of the triangle.

  1. $18$
  2. $23$
  3. $22$
  4. $21$
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D Correct answer
Explanation

Using the formula for the length of an internal angle bisector d = 2ab cos(C/2) / (a+b), where a=9, b=18, d=8. This leads to cos(C/2) = 8 * 27 / (2 * 9 * 18) = 216 / 324 = 2/3. Using the law of cosines, c^2 = a^2 + b^2 - 2ab cos(C). Since cos(C) = 2cos^2(C/2) - 1 = 2(4/9) - 1 = -1/9, c^2 = 81 + 324 - 2(9)(18)(-1/9) = 405 + 36 = 441. c = 21.

AI explanation

Let the longest side be divided into lengths m and n by the angle bisector; using the angle bisector proportionality theorem, we establish the ratio of the sides as 9 divided by 18, meaning m divided by n equals 1 divided by 2. Applying the angle bisector length formula, 8 squared equals 9 times 18 minus m times n, which simplifies to 64 equals 162 minus m times n. This yields a product for m times n equal to 98, and substituting m equals n divided by 2 gives n squared equals 196. Solving for the lengths gives m as 7 and n as 14, so adding them together makes the longest side 21.

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