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
