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 =p/q , where p and q are co-prime, then find the sum of digits of p + q.
Answers
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
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