In the δabc, the altitude an = 24 in, bn = 18 in, ac = 40 in. find ab and bc.

AB = 30 in and BC = 50 in.

We use Pythagorean theorem to solve this.  Since AN is an altitude, this means that it is perpendicular to BC.  This means BN and AN are the legs of one right triangle, with AB being the hypotenuse:

18²+24² = AB²
324 + 576 = AB²
900 = AB²

Take the square root of both sides:
√900 = √AB²
30 = AB

NC and AN form the legs of the other right triangle, with AC being the hypotenuse:

24²+NC² = 40²
576 + NC² = 1600

Subtract 576 from both sides:
576 + NC² - 576 = 1600 - 576
NC² = 1024

Take the square root of both sides:
√NC² = √1024
NC = 32

BC = BN + NC = 18 + 32 = 50