In triangle ABC, the angle ABC = 90 degrees, and BH is an altitude. Find segment BH if segment AH = 4 and segment HC = 1.

Answer 1

Since BH is an altitude of triangle ABC, it divides the triangle into two right triangles, triangle ABH and triangle CBH.

Using the Pythagorean theorem in triangle ABH, we have:

BH^2 + AH^2 = AB^2
BH^2 + 4^2 = AB^2
BH^2 + 16 = AB^2

Similarly, in triangle CBH, we have:
BH^2 + HC^2 = CB^2
BH^2 + 1^2 = CB^2
BH^2 + 1 = CB^2

Since angle ABC = 90 degrees, AB is the hypotenuse of triangle ABH and CB is the hypotenuse of triangle CBH, which means AB = CB.

So, we have:

BH^2 + 16 = AB^2
BH^2 + 16 = CB^2
BH^2 + 16 = BH^2 + 1

Subtracting BH^2 from both sides:

16 = 1

This is a contradiction, so there must be a mistake in the given information.