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Olympiad workbookIOQMProperties of Triangles

Let A B C be a triangle with A B=A C . Let D be a point on the segment B C such that B D=48 1 61 and D C=61 . Let E be a point on A D such that C E is perpendicular to A D and D E=11 . Find A E .

Correct answer

25

Step-by-step solution

Let B D=48 1 61 = , M is mid-point of B C . aligned & C D=61= & B M=C M= + 2 & D M=B M-B D= - 2 aligned Let A E=x, A B=A C=y, A M=h, C E=z In C D E , aligned & C E^2=C D^2-D E^2=61^2-11^2=50 72 & C E=60=z ...(i) & In A C E, y^2=x^2+60^2 & y^2=x^2+3600 ...(ii) aligned In A C M, y^2=h^2+ ( + 2 )^2 ...(iii) In A D M,(x+11)^2=h^2+ ( - 2 )^2 ...(iv) (ii) & (iii) x^2+3600=h^2+ ( + 2 )^2 ...(v) aligned & (iv) &( v ) (x+11)^2-x^2-3600= ( - 2 )^2- ( + 2 )^2 & 22 x+121-3600=- & =- (48 61)+1 61 61 & 22 x=3600-121-2929 & x=25=

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