KatherineGrell

To find the lengths BC and BZ in triangle AB, we can use the angle bisector theorem.

The angle bisector theorem states that in a triangle, if an angle bisector divides the opposite side into two segments, the ratio of the lengths of those segments is equal to the ratio of the lengths of the other two sides.

Let's use the angle bisector theorem to find BC and BZ.

First, let's consider the angle bisector BY. According to the angle bisector theorem, we have:

AY / CY = AB / BC

Substituting the given values, we get:

16 / 16 = 16 / BC

Simplifying the equation, we have:

1 = 16 / BC

Cross-multiplying, we get:

BC = 16

So, BC has a length of 16.

Next, let's consider the angle bisector CZ. According to the angle bisector theorem, we have:

AY / BY = AC / BC

Substituting the given values, we get:

16 / BY = (16 + BC) / BC

Since we already found that BC = 16, we can substitute that value:

16 / BY = (16 + 16) / 16

Simplifying the equation, we have:

16 / BY = 2

Cross-multiplying, we get:

BY = 8

Now, we have the length of BY, but we want to find BZ. Since BY is an angle bisector, it divides the angle at B into two equal angles. This means that triangle BYZ is isosceles, and BZ has the same length as BY.

Therefore, BZ = BY = 8.

In summary, in triangle AB, the length of BC is 16, and the length of BZ is 8.