Geometry and Trigonometry practice question

Answer: 60

The altitude to the hypotenuse creates triangles ABD, BCD, and ABC that are all similar. From triangle ABD ~ triangle BCD, AD/BD = BD/DC, so 9/12 = 12/DC and DC = 144/9 = 16. Then AC = 9 + 16 = 25. Using the right triangles ABD and BCD: AB = √(9² + 12²) = 15 and BC = √(12² + 16²) = 20. Perimeter = 15 + 20 + 25 = 60. (Check: 15² + 20² = 625 = 25².) Common wrong results: 46 comes from adding 9 + 12 + 25 after assuming BD is a side of triangle ABC; 48 comes from assuming D is the midpoint of AC (DC = 9), giving AC = 18 and AB = BC = 15; 57 comes from taking DC = 12 (mistaking the altitude for the second hypotenuse segment), giving AC = 21 and BC = 12√2 rounded. Only the geometric-mean relation DC = BD²/AD yields the correct segments.

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