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AB is a diameter of a circle. From a, two secants AP and AQ have been drawn which intersect the tangent at B to the circle at points L and M respectively. Prove that:- triangle APQ is similar to traingle AML and P,Q,M,L are concyclic
Asked by rushabhjain.avv | 30 Jan, 2019, 08:20: PM
answered-by-expert Expert Answer
 
Figure shows a circle with diameter AB. A tangent is drawn at B. Secants AP and AQ intersect the tangenet at L and M.
 
By tangent secant theorem, we have LB2 = LA×LP  = LA×(LA-AP) = LA2 - LA×AP ..............(1)
 eqn.(1) is written as,   LA × AP =  LA2 - LB2 = AB2 ...............................(2)
 
Similarly, by tangent secant theorem, MB2 = MA×MQ = MA×(MA-AQ) = MA2 - MA×AQ ................(3)
eqn.(3) is written as,   MA × AQ = MA2 - MB2 = AB2 .........................(4)
 
from (2) and (4), we have,   LA × AP = MA × AQ or begin mathsize 12px style fraction numerator L A over denominator A Q end fraction space equals space fraction numerator M A over denominator A P end fraction end style  ............................(4)
In ΔAPQ and ΔAML, begin mathsize 12px style angle end stylePAQ is common angle and sides of the common angles  are in a ratio given by eqn.(4).
Hence ΔAPQ is similar to ΔAML
 
in similar triangles ΔAPQ and ΔAML , we have begin mathsize 12px style angle end styleAPQ = begin mathsize 12px style angle end styleAML ...................(5)
also begin mathsize 12px style angle end styleAQP = begin mathsize 12px style angle end styleALM  ......................(6)
since begin mathsize 12px style angle end styleAPQ + begin mathsize 12px style angle end styleLPQ = 180, using eqn.(5 ), begin mathsize 12px style angle end styleLPQ+begin mathsize 12px style angle end styleQML = 180 .....................................( 7)
similarly, since begin mathsize 12px style angle end styleAQP + begin mathsize 12px style angle end stylePQM = 180, using eqn.( 6), begin mathsize 12px style angle end stylePQM + begin mathsize 12px style angle end stylePLM = 180 .................(8)
from (7) and (8 ), we have, PQML is cyclic quadrilateral and points P, Q, M and L are concyclic
Answered by Thiyagarajan K | 01 Feb, 2019, 08:46: AM

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