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ICSE Class 10 Answered

The two medians BE andCF of triangle ABC intersect eachother at G.If the line segment FE intersect the line segment  AG at O, then let us prove that AO=3 OG.
Asked by naskar.koushik94 | 27 Jun, 2019, 10:26: AM
answered-by-expert Expert Answer
Extend AO to meet BC at D
G is the centroid of triangle ABC
D, E and F are the mid-points of BC, AC and AB 
so, O is the midpoint of EF
fraction numerator E O over denominator O F end fraction equals 1 half T h e r e f o r e comma space fraction numerator O F over denominator B D end fraction equals 1 half A l s o comma space fraction numerator begin display style O F end style over denominator begin display style B D end style end fraction equals fraction numerator A O over denominator A D end fraction rightwards double arrow fraction numerator begin display style A O end style over denominator begin display style A D end style end fraction equals fraction numerator begin display style 1 end style over denominator begin display style 2 end style end fraction rightwards double arrow A D equals 2 A O rightwards double arrow A O equals O D N o w comma space sin c e space G space i s space t h e space c e n t r o i d rightwards double arrow fraction numerator A G over denominator G D end fraction equals 2 over 1 rightwards double arrow fraction numerator begin display style A O plus O G end style over denominator begin display style O D minus O G end style end fraction equals fraction numerator begin display style 2 end style over denominator begin display style 1 end style end fraction rightwards double arrow A O plus O G equals 2 O D minus 2 O G rightwards double arrow 3 O G equals 2 O D minus A O rightwards double arrow 3 O G equals A O rightwards double arrow A O equals 3 O G
Answered by Renu Varma | 28 Jun, 2019, 10:25: AM

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