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Geometry Level 4

In the figure above, area of circle is 50 and area of triangle is 15.

If the value of sin ⁡ θ + sin ⁡ α + sin ⁡ β \sin\theta + \sin\alpha + \sin\beta equal to m n π \dfrac mn \pi for coprime positive integers m m and n n , find the value of m + n m+n .


The answer is 8.

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Sakanksha Deo by Sakanksha Deo , 23, India

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1 solution

Sakanksha Deo
Mar 9, 2015

Let the radius of the circle be r,

Area of cilcle = 50 = π r 2 50 = \pi r^{2} ...... (1)

Now,

Ar( PQR ) = ar( OPQ ) + ar( OQR ) ar( OPR )

⇒ 15 = 1 2 r 2 sin ⁡ α + 1 2 r 2 sin ⁡ θ + 1 2 r 2 sin ⁡ β \Rightarrow 15 = \frac{1}{2} r^{2} \sin\alpha + \frac{1}{2} r^{2} \sin\theta + \frac{1}{2} r^{2} \sin\beta

⇒ ( sin ⁡ α + sin ⁡ β + sin ⁡ θ ) = 30 r 2 \Rightarrow (\sin\alpha + \sin\beta + \sin\theta )= \frac{30}{ r^{2}}

From (1),

⇒ sin ⁡ α + sin ⁡ θ + sin ⁡ β = 3 π 5 = 3 π 5 \Rightarrow \sin\alpha + \sin\theta + \sin\beta = \frac{3 \pi }{5} = \frac{3 \pi }{5}

m + n = 8 \boxed{8}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Mine too was a similar approach.. :)

Rishabh Tripathi - 6 years, 2 months ago

I loved the problem sir.

Sathvik Acharya - 4 years, 2 months ago

Can you please tell me the source?

Sathvik Acharya - 4 years, 2 months ago

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