Question

ABC is a right triangle. If AB = 3 and AC = 7, find BC. Leave your answer in simplest radical form.

Answer 1

1. Suppose that BC is hypotenuse, then by Pythagorean theorem:
$BC^2=AC^2+AB^2, \\ BC^2=3^2+7^2, \\ BC^2=9+49, \\ BC^2=58, \\ BC= \sqrt{58}$.
2. If BC is not hypotenuse, then BC is a leg and AC is hypotenuse (because AC>AB). Hence by Pythagorean theorem:
$AC^2=AB^2+BC^2, \\ 7^2=3^2+BC^2, \\ BC^2=49-9, \\ BC^2=40, \\ BC= \sqrt{40} =2 \sqrt{x10}$.

Answer 2

If AB is the shortest side and AC is not the hypotenuse the answer would be √58.
If AC is the hypotenuse the answer would be 2√10.