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help me in this geometry question?
On triangle ABC, points D and E are chosen on line segments AB and AC, respectively. The lines CD and BE intersect at K, within triangle ABC. If [DKB]=56,[BKC]=84 and [KEC]=90, what is [ADKE]?
Note: [PQRS] denotes the area of the figure PQRS.
1 Answer
- DukeLv 79 years agoFavorite Answer
Let XYZ is arbitrary triangle, T is a point on the segment XY, then in your notations
[XZT] : [YZT] = |XT| : |TY|
because both triangles XZT and YZT have common altitude through Z.
Now your problem. Connect D with E.
[BDK] : [EDK] = |BK| : |EK| = [BCK] : [ECK] = 84 : 90, hence
[EDK] = [BDK] * 90/84 = 60
Next [ADE] : [BDE] = |AD| : |BD| = [ACD] : [BCD] or
[ADE] : (56 + 60) = ([ADE] + 60 + 90) : (84 + 56)
Solving this equation you get [ADE] = 725 and [ADKE] = 725 + 60 = 785.
