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CBSE Class 9 Answered

In ∆ ABC, the bisectors of ∟B and ∟C intersects at I. A line drawn through I and parqallel to BC intersect AB at P and AC at Q. Prove that PQ = BP + CQ.
Asked by suvi7726 | 29 Jan, 2019, 10:46: PM
answered-by-expert Expert Answer

Figure shows the angular bisectors of begin mathsize 12px style angle end styleABC and begin mathsize 12px style angle end styleACB meet at I in a ΔABC. Line LIQ is parallel to BC.
begin mathsize 12px style angle end stylePIB = begin mathsize 12px style angle end styleIBC ...............(1) 
we write eqn.(1) because both angles are alternate angles between parallel lines which is intercepted by IB
 
we are given that IB is angular bisector, hence begin mathsize 12px style angle end stylePBI = begin mathsize 12px style angle end styleIBC  .................(2)
from (1) and (2), begin mathsize 12px style angle end stylePBI = begin mathsize 12px style angle end stylePIB , hence ΔPBI is isoceless triangle and PB = PI  ..................(3)
Similarly we can show that, QC = QI ......................(4)

adding eqns.(3) and (4), we get PB+QC = PI+QI = PQ
Answered by Thiyagarajan K | 30 Jan, 2019, 10:27: AM

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