Three points have coordinates A(6,6), B(-3,3) and C(9,k). The foot of the perpendicular from A to BC is the midpoint of BC. Calculate the possible values of k?

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Answer 1 · 2018-02-28T03:11:56

#color (green)(k = 15, -3)#

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Given D is the mid point of BC and AD is also perpendicular to BC.

Hence, #AD# is perpendicular bisector of side #BC#.

Or, #AC = AB#. I.e., ABC is an isosceles triangle.

#vec(AC)^2 = (9-6)^2 +( k-6)^2=> 9 + (k-6)^2#

#vec(AB)^2 = (6+3)^2 + (6-3)^2 = 90#

Since #vec(AB) = vec (AC)#,

#(9 + (k-6)^2 )= 90#

#(k-6)^2 = 90 - 9 = 81#

#k - 6 = +-9# or k = 15, -3#

Point C (9, 15) or (9, -3)#