Question

A triangle has corners at points A, B, and C. Side AB has a length of `7`. The distance between the intersection of point A's angle bisector with side BC and point B is `4`. If side AC has a length of `14`, what is the length of side BC?

Answer 1

`BC=12`

Explanation:

`" let D be the point on BC where the angle bisector"`
`"meets BC"`

`"using the "color(blue)"angle bisector theorem"`

`color(red)(bar(ul(|color(white)(2/2)color(black)((AC)/(AB)=(CD)/(BD))color(white)(2/2)|)))`
require to find CD

`rArr14/7=(CD)/4larrcolor(blue)" cross-multiply"`

`rArr7CD=4xx14`

`rArrCD=(4xx14)/7=8`

`rArrBC=CD+BD=8+4=12`