# maths

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If ABC is a triangle with AB=20,BC=22 and CA=24. Let D lie on BC such that AD is the angle bisector of ∠BAC. What is AD2?

• maths -

I assume the last sentence is

Hints:

Use cosine rule to find ∠A, in triangle ABC.
A=arccos((20²+24²-22²)/(2*20*24))
=arccos(41/80)
= 59° (approx.)

Then
Area of triangle ABC
=(1/2)AB.AC.sin(A)
=(1/2)*20*24*(11√39)/80
=33√39

Since AD is angle bisector, the altitudes of D to AB equals that to AC.
Therefore AD divides the area of triangle ABC in ratios of the bases AB and AC. Thus
=18√39

Area of ΔADC can also be found by the trigonometric formula

=>
18√39 = AD * 24 *sin(A/2)
=>

• maths -

• maths -

The solution is given for AD.
Are you able to calculate AD²?

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