What Center Is Most Crucial?

Geometry Level 4

A triangle P Q R PQR was drawn on a Cartesian plane such that there exists a point O O for which the distance P O ‾ = Q O ‾ = R O ‾ \overline {PO } = \overline{QO} = \overline{RO} is equal to 4. And two of the vertices of this triangle have coordinates ( 1 , 3 ) (1,3) and ( 5 , 6 ) (5,6) .

Given that the sine of one of the interior angles of this triangle must always be a constant. Find this constant.

Give your answer to 3 decimal places.


The answer is 0.625.

Pi Han Goh by Pi Han Goh , 30, Malaysia

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2 solutions

Point O O appears to be the circumcentre, and the triangle P Q R PQR has circumradius 4. Applying the sine rule, we find that p sin ⁡ P = 2 r ⇒ 5 sin ⁡ P = 8 ⇒ sin ⁡ P = 5 8 = 0.625 \frac{p}{\sin P}=2r\Rightarrow \frac{5}{\sin P}=8\Rightarrow \sin P=\frac{5}{8}=0.625

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Ahmad Saad
Jun 19, 2016

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