Question

Stuck on this question... solve using the angle bisector theorem and use X to find BC (30 pts)

Answer 1

since AD is an angle bisector to ∡BAC, then by the angle bisector, triangles ABD ~ ADC, therefore, we can use their corresponding sides, thus

`\bf \cfrac{AB}{AC}=\cfrac{BD}{DC}\implies \cfrac{8}{12}=\cfrac{x+1}{2x}\implies \cfrac{2}{3}=\cfrac{x+1}{2x}\implies 4x=3x+3\implies \boxed{x=3} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ BC=(x+1)+(2x)\implies BC=4+6\implies BC=10`

Answer 2

The angle bisector theorem tells you the segments on one side of the bisector are proportional to those on the other side. This lets you write

`(x+1)/(8)=(2x)/(12)\\\\3(x+1)=2(2x)\qquad\text{multiply by 24}\\\\3x+3=4x\qquad\text{eliminate parentheses}\\\\x=3\qquad\text{subtract 3x}\\\\BC=(x+1)+(2x)=(3+1)+(2\cdot 3)\\\\BC=4+6=10`

The length of BC is 10.