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

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

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    2021-08-13T05:01:46+00:00

    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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