AAS congruency doubt
this question is from XAT 2009.I am having doubt in it regarding AAS congruency, i have uploaded question fig. also
Q)ABCD is a square with sides of length 10 units. OCD is an isoceles triangle with base CD. OC cuts AB at pt Q and OD cuts AB at point P. The area of trapezoid PQCD is 80 square units.The altitude from O of the triangle OPQ is :
(a) 12 (b)13 (c)14 (d)15 (e) None of these.
Now my approach using AAS congruency was
area of trapezium ABCD = 80 (given), 80 = 1/2 x (PQ+CD) x AC
80 = 1/2 x (PQ+10) x 10
therfore PQ = 6
now if i prove triangles OQR & ORP CONGRUENT to each other by AAS rule (angle OAR = angle OPR; as both triangles are isosceles ; angle ORA=angle ORP= 90 ; OR= common side) now as both the triangles are congruent therefore QR=RP = 3 (since congruent triangles have equal corresponding sides)
triangles OQR and triangles ORP are also similar by AA similarity ; therfore QR/CS = OR/CS
3/5 = OR/10 => OR = 6
Now OS= OR+RP = 6+10 = 16 units
But unfortunately the correct answer is 15 units Hence i want to know whether AAS Congruency i used here is wrong ; But as far as i know all maths textbooks have proved AAS congruency.
Re: AAS congruency doubt
hey karan, ur approach is correct but the calculation is incorrect. while applying AA similarity, u have considered OR/CS which is completely incorrect. it should be OR/OS.|
now, u can take OS=OR+10. then solve. u'll get the correct answer