## In ΔABC, median AD on BC and the angle bisector BE (where E lies on AC) are perpendicular to each other. If AD = 5 cm and BE = 7 cm and area

Question

In ΔABC, median AD on BC and the angle bisector BE (where E lies on AC) are perpendicular to each other. If AD = 5 cm and BE = 7 cm and area of ΔABC = , where p and q are co-prime, then find the sum of digits of p + q.

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1 month 2021-08-13T05:00:43+00:00 1 Answer 0 views 0

1. Given :  ΔABC, median AD on BC and the angle bisector BE (where E lies on AC) are perpendicular to each other. If AD = 5 cm and BE = 7 cm and area of ΔABC =p/q , where p and q are co-prime

To find  : sum of digits of p + q.

Solution :

Let say BE intersect AD at X

then BX is angle bisector of ∠B in Δ ABD

=> ∠XBA = ∠ XBD

∠BXA = ∠BXD = 90°  as   BE ⊥ AD

BX = BX

=> Δ BXD ≅ Δ BXA

=> AX = XD     &   AB = BD

AX + XD = AD = 5

=> AX =  5/2  cm

Area of Δ ABE =  (1/2)BE * AX

= (1/2) * 7 * (5/2)

= 35/4  cm²

Area of Δ ABE = 35/4  cm²

AB = BD

BC = 2 BD    = 2 AB

now BE is angle bisector of Δ ABC

=>  AB/AE = BC/CE

=> AB / AE = 2AB / CE

=> CE = 2AE

=> AE  =  AC/3

=> Area of Δ ABC = 3 area of Δ ABE

=> Area of Δ ABC =   3 ( 35/4)   cm²

=>  Area of Δ ABC =   105/4   cm²

p = 105

q = 4

p + q  = 109

Sum of digits of of  109 = 1 + 0 + 9 = 10