Areas and Ratios

Geometry Level 2

Given below is an isosceles triangle with A B = B C AB = BC . X Y XY is a line parallel to A C AC such that it divides triangle A B C ABC into two parts of equal area. Find the ratio of the length of A X AX to that of A B AB .

BONUS : Is the ratio same for every possible triangle?

( 2 − 2 ) : 2 (\sqrt{2} - 2) : 2 ( 2 − 1 ) : 2 (\sqrt{2} - 1) : 2 ( 2 − 1 ) : 2 (\sqrt{2} - 1) : \sqrt{2} ( 2 − 2 ) : 2 (\sqrt{2} - 2) : \sqrt{2}

Abha Vishwakarma by Abha Vishwakarma , 20, India

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1 solution

Aryan Gupta
Oct 3, 2018

OUTLINE OF SOLN.

ar(XYB):ar(ABC)=1:2

therefore, XB:AB=1:root2 (by basic similarity applications)

On rearranging you get the answer.

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