In triangle AB, let the angle bisectors be BY and CZ. Given AB = 16, AY = 16, and CY = 16, find BC and BZ.
We know that the angle bisector theorem and the specified requirements are satisfied if BC = 16, CZ = 8, and BZ = 24. The angle bisector theorem is also satisfied if BC = 8, which also makes CZ = 16/3 and BZ = 112/9.
What is angle bisector theorem?
The opposite side of a triangle is divided into two portions that are proportional to the sides they are on if the angle is bisected by a line, according to the angle bisector theorem.
The angle bisector theorem, which asserts that if a line bisects a triangle's angle, it divides the opposite side into two segments that are next sides to each other, can be used to solve this problem.
Let BC and BZ both equal x. The angle bisector theorem provides us with:
AC/AB = CZ/BZ
16/(16+x) = 16/y
We solve y in terms of x, and as a result,
y = 16(16+x)/16 = x+16
Upon substitution, we discover—
16/(16+x) = 16/(x+16)
If we cross-multiply, we obtain:
16(x+16) = 16(16+x)
Simplifying, we get:
16x + 256 = 256 + 16x
In light of the fact that 16x cancels on both sides, we are left with:
256 = 256
As a result, there exist an endless number of BC and BZ values that could satisfy the requirements.