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Olympiad workbookIOQMProperties of Triangles

In a triangle A B C , the median A D divides B A C in the ratio 1: 2 . Extend A D to E such that E B is perpendicular A B . Given that B E=3, B A=4 , find the integer nearest to B C^2 .

Correct answer

29

Step-by-step solution

Here, D is mid-point of B C , hence B D: C D=1: 1 A B E=90^ Let B A D= , then C A D=2 = 3 4 , and 2 = 24 7 Now, using m-n theorem in A B C aligned 2 & = - 2 & = 25 48 aligned Now, using sine rule in A B D , we get aligned & B D 4 = ( - ) & B D= 4 3 25^2+48^2 5 48 aligned So, 4 B D^2=B C^2= 25^2+48^2 100 =29.29 Nearest integer is 29 .

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