Answer:
AE=5 and EC=10
Step-by-step explanation:
Given : Suppose BE is an angle bisector of ΔABC , AB=9, BC=15, and AC=16
To find : AE and EC
Solution : BE form an angle bisector on line AC
Let AE=x and EC = y
Length of AC= AE+EC , putting values
16=x+y .....[1]
Now, we apply angle bisecor theorem which state that the ratio of the length of the line segment AE to the length of segment EC is equal to the ratio of the length of side AB to the length of side AC
[tex]\frac{|AE|}{|EC|}=\frac{|AB|}{|AC|}[/tex]
Put values,
[tex]\frac{|x|}{|y|}=\frac{|9|}{|16|}[/tex]
Cross multiply,
[tex]\rightarrow 15x=9y[/tex]
[tex]\rightarrow 15x-9y=0[/tex].......[2]
From [1] and [2]
Substitute x from [1] in [2], x= 16-y
[tex]15(16-y)-9y=0[/tex]
[tex]\rightarrow 240-15y-9y=0[/tex]
[tex]\rightarrow 240=24y[/tex]
[tex]\rightarrow y=10[/tex]
Put y in value of x,
x= 16-y
x=16-10=6
Therefore, length of AE=x=5 and EC=y=10
