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Olympiad workbookIOQMProperties of Triangles

In a triangle ABC , the median AD (with D on BC ) and the angle bisector BE (with E on AC ) are perpendicular to each other. If AD =7 and BE =9 , find the integer nearest to the area of triangle A B C .

Correct answer

47

Step-by-step solution

Let A D and B E meet at F Now ABF = FBD = B 2 and AFB = BFD =30^ and BF is common to triangles ABF and BFD hence the two triangles are Cengruent so AF = FD = 7 2 Also, AB = BD , AB : BC =1: 2 AE : EC =1: 2 So area of triangle A B C=3 (Area of A B E ) aligned & =3 ( 1 2 AF BE ) & = 3 2 7 2 9= 189 4 =47.25 aligned Hence nearest integer = 47

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