and are angle bisectors of triangle ABC that meet at I as shown below, with CY = 4, AY = 6, and AB = 8. Find BZ.

The length of the segment  BZ is approximately 2.78

The above value is obtained by using the following process:

The given parameters of the triangle are;

The angle bisectors of the triangle = and

The point at which and meet = The point I

The length of CY = 4

The length of AY = 6

The length of AB = 8

The unknown parameter;

The length of the segment BZ

Method:

Apply the angle bisector theorem

According to the angle bisector theorem we have;

,

From ,  we get;

6 × BC = 8 × 4 = 32

BC = 32/6 = 16/3 =  

BC =  

From , we have;

AC = CY + AY

∴ AC = 4 + 6 = 10

AZ = AB - BZ

∴ AZ = 8 - BZ

Therefore;

10 × BZ = ×(8 - BZ) = (16/3) × 8 -  (16/3)·BZ

10·BZ + (16/3)·BZ = (16/3) × 8

(46/3)·BZ = 128/3

BZ = 128/46 = 64/23

The length of the line BZ = 64/23 = 2≈ 2.78

Learn more about angle bisector theorems here;

link

  BZ = 64/23 = 2 18/23

Step-by-step explanation:

The angle bisectors divide the triangle into proportional parts, so we have ...

  CB/CY = AB/AY

  CB = CY(AB/AY) = 4(8/6) = 16/3 . . . . multiply by CY and evaluate

and ...

  BZ/CB = AZ/AC

  BZ = AZ(CB/AC) . . . . multiply by CB

But we have ...

  AZ = 8 -BZ, so this becomes ...

  BZ = (8 -BZ)(16/3)/(10)

  BZ(1 +8/15) = 64/15 . . . . add the BZ term on the right to both sides

  BZ = (64/15)(15/23) = 64/23 . . . . divide by the coefficient of BZ

BZ = 64/23 = 2 18/23

333.74 ! should be your answer for this question