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Circumscribed circle of the hexagon | PRMO 2018 | Problem 7

Try this beautiful problem from Geometry based on Radius of the circumscribed circle of the hexagon .You may use sequential hints to solve the problem

Try this beautiful problem from Geometry based on Radius of the circumscribed circle of the hexagon

Hexagon|PRMO| 2007


A point P in the interior of a regular hexagon is at distance 8,8,16 units from three consecutive vertices of the hexagon, respectively. If r is radius of the circumscribed circle of the hexagon, what is the integer closest to r ?

Circumscribed circle of the hexagon
  • $12$
  • $14$
  • $16$

Key Concepts


Geometry

Triangle

Hexagon

Check the Answer


But try the problem first…

Answer:$14$

Source
Suggested Reading

PRMO (2018) Problem 7

Pre College Mathematics

Try with Hints


First hint

Show \(\triangle PAB \)is similar to \(\triangle PFC \).

Can you now finish the problem ……….

Second Hint

The circumradius of a regular hexagon = side of regular hexagon

can you finish the problem……..

Final Step

finding radius of Circumscribed circle of the hexagon

Note that CF = 2AB, PA = 2PC & PB = 2PF PFC, hence \(\triangle PAB \)is similar to \(\triangle PFC \).Hence \(A_1P_1C  and B_1P_1F\)  are collinear. Let each side of hexagon be equal to x . Let Q & R be foot of altitudes from P to base AB & CF respectively. So R is centre of hexagon

Now

\(\frac{1}{3} \times \frac {\sqrt 3 x}{2}\) =\(\sqrt{64-\frac{x^2}{4}}\)

\(\Rightarrow 64-\frac{x^2}{4}\)

\(\Rightarrow \frac{4 x^2}{12} =64\)

\(\Rightarrow x=8\sqrt 3\)

We know that the circumradius of a regular hexagon = side of regular hexagon

Hence r=8\(\sqrt 3\)\(\approx\) 13.856

Therefore r=14 ( nearest integer)

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