Pre-RMO 2014/10

In a triangle $$ABC$$, $$X$$ and $$Y$$ are points on the segments $$AB$$ and $$AC$$, respectively, such that $$AX : XB = 1 : 2$$ and $$AY : YC = 2 : 1$$. If area of triangle $$AXY$$ is $$10$$ then what is the area of triangle $$ABC$$?

This note is part of the set Pre-RMO 2014

Note by Pranshu Gaba
3 years, 8 months ago

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- 2 years, 9 months ago

Let side $$AB = c$$, $$AC = b$$, and $$\angle BAC = \theta$$. Then $$AX = \frac{c}{3}$$ and $$AY = \frac{2b}{3}$$.

Area of triangle $$ABC = \frac{1}{2} bc \sin\theta$$ and area of triangle $$AXY$$ = $$\frac{1}{2} \times \frac{2b}{3} \times \frac{c}{3} \sin \theta = 10$$

$\text{Area }\triangle AXY = \dfrac{2}{9}( \text{Area }\triangle ABC)$

$\text{Area }\triangle ABC = \dfrac{9}{2} ( \text{Area }\triangle AXY)$

$\text{Area }\triangle ABC = \dfrac{9}{2} \times 10 = \boxed{45}$

- 3 years, 8 months ago

- 3 years, 8 months ago

90??

- 3 years, 8 months ago