How would you prove that two altitudes of a scalene triangle are not congruent?

Answer 1

1. Draw scalene triangle ABC.
2. Draw altitudes BM and CN (M is on AC, N is on AB).

Taking liberties on notation BM = "length of BM", and similar for other segments. A more formal proof would use |BM|, |CN|, etc.

1. Proof by Contradiction: Assume BM = CN.
2. For ΔBNC: BC² = CN² + BN²
3. For ΔBMC: BC² = CM² + BM²
4. So we can say (remember BM = CN) BN = CM
5. Repeat for ΔANC ( AC² = CN² + AN² ) and ΔAMB ( AB² = BM² + AM² )

....You will be able to show this leads to AN = AM

This result, plus the result on line 6 (BN = CM) leads to a contradiction (namely AC = AB which contradicts the premise that the triangle is scalene).

Answer 2

One way is via trig....

Dropping height from C, sin A = H1/B

Dropping height from A, sin B = H2/C

since the sides of the scalene triangle are

different, the angles and trig ratios also MUST differ,

which in turn forces H1 and H2 to differ