One angle of a triangle is equal to one angle of another triangle and the bisectors of these equal angles divide the opposite sides in the same ratio. Prove that the triangles are similar.( I want to know how this id directly assumed during the proof: the ratio of the corresponding sides=the ratio of the bisected side)

Asked by sajal2402 | 12th Feb, 2019, 10:36: AM

Expert Answer:

Angle bisector theorem :- anglar bisector of an angle of triangle divides the opposite side in same ratio as their sides of the smae angle.
 
In ΔABC, AD is bisector of begin mathsize 12px style angle end styleB , hence begin mathsize 12px style fraction numerator B D over denominator D C end fraction equals space fraction numerator A B over denominator A C end fraction end style
In the figure, In triangles ABC and PQR, begin mathsize 12px style angle end styleB = begin mathsize 12px style angle end styleP .  AD and PS are respective angular bisector.
then we have begin mathsize 12px style fraction numerator B D over denominator D C end fraction equals space fraction numerator A B over denominator A C end fraction end style  ...........................(1)
begin mathsize 12px style fraction numerator Q S over denominator S R end fraction space equals space fraction numerator P Q over denominator P R end fraction end style .............................(2)
Given : bisectors of begin mathsize 12px style angle end styleB and begin mathsize 12px style angle end styleP divides the opposite side in same ratio,  hence begin mathsize 12px style fraction numerator B D over denominator D C end fraction equals space fraction numerator Q S over denominator S R end fraction end style .........................(3)
By comparing (1), (2) and (3), we have begin mathsize 12px style fraction numerator A B over denominator A C end fraction equals fraction numerator P Q over denominator P R end fraction end style..........................(4)
In ΔABC and ΔPQR, begin mathsize 12px style angle end styleB = begin mathsize 12px style angle end styleP and the sides of these equal angles are in same ratio.  Hence ΔABC and ΔPQR are similar triangles

Answered by Thiyagarajan K | 12th Feb, 2019, 11:17: AM

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