Please explain the exterior angle bisector theorem!!!
Exterior angle bisector theorem : The external bisector of an angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle. produced in D.Exterior Angle Bisector Theorem
Given : A ΔABC, in which AD is the bisector of the exterior ∠A and intersects BC
Prove that : BD / CD = AB / AC
Construction : Draw CE || DA meeting AB in E.
1) CE || DA 1) By construction 2) ∠1 = ∠3 2) Alternate interior angle 3) ∠2 = ∠4 3) Corresponding angle (CE ||DA and BK is a transversal 4) AD is a bisector of ∠A 4) Given 5) ∠1 = ∠2 5) Definition of angle bisector 6) ∠3 = ∠4 6) Transitivity (from 2 and 4) 7) AE = AC 7) If angles are equal then side opposite to them are also equal 8) BD / CD = BA/EA 8) By Basic proportionality theorem(EC ||AD) 9) BD /CD = AB/AE 9) BA = AB and EA = AE 10) BD /CD = AB /AC 10) AE = EC and from(7)
Examples
1) In the given figure, AE is the bisector of the exterior ∠CAD meeting BC produced in E. If AB = 10 cm, AC = 6 cm and BC = 12 cm, find CE.
Given : AB = 10 cm, AC = 6 cm and BC = 12 cm
By exterior angle bisector theorem
BE / CE = AB / AC
(12 + x) / x = 10 / 6
6( 12 + x ) = 10 x [ by cross multiplication]
72 + 6x = 10x
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72 = 10x – 6x
72 = 4x
x = 72/4
x = 18
CE = 18 cm
PLZ...... GIVE THUMPS UP..
