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 = [tex]\mathbf{\overline{BY}}[/tex] and [tex]\mathbf{\overline{CZ}}[/tex]
The point at which [tex]\overline{BY}[/tex] and [tex]\overline{CZ}[/tex] 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;
[tex]\mathbf{\dfrac{AY}{CY} = \dfrac{AB}{BC}}[/tex], [tex]\mathbf{\dfrac{BZ}{AZ} = \dfrac{BC}{AC}}[/tex]
From [tex]\mathbf{\dfrac{AY}{CY} = \dfrac{AB}{BC}}[/tex], we get;
[tex]\dfrac{6}{4} = \dfrac{8}{BC}[/tex]
6 × BC = 8 × 4 = 32
BC = 32/6 = 16/3 = [tex]5\frac{1}{3}[/tex]
BC = [tex]5\frac{1}{3}[/tex]
From [tex]\mathbf{\dfrac{BZ}{AZ} = \dfrac{BC}{AC}}[/tex], we have;
AC = CY + AY
∴ AC = 4 + 6 = 10
AZ = AB - BZ
∴ AZ = 8 - BZ
Therefore;
[tex]\mathbf{\dfrac{BZ}{8 - BZ} = \dfrac{5\frac{1}{3} }{10}}[/tex]
10 × BZ = [tex]5\frac{1}{3}[/tex] ×(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[tex]\frac{18}{23}[/tex] ≈ 2.78
Learn more about angle bisector theorems here;
https://brainly.com/question/8741067
