Try this beautiful problem from the Pre-RMO, 2019 based on Diameter of a circle.
A village has a circular wall around it, and the wall has four gates pointing north, southeast and west. A tree stands outside the village, 16 m north of the north gate, and it can be just seen appearing on the horizon from a point 48 m east of the south gate. Find the diameter in meters of the wall that surrounds the village.
Pythagoras Theorem
Equations
Integer
But try the problem first...
Answer: is 48.
PRMO, 2019, Question 25
Geometry Vol I to IV by Hall and Stevens
First hint
Let radius =r
or,\(AB=\sqrt{AO^{2}-OB^{2}}=\sqrt{(16+r)^{2}-r^{2}}\)
\(=\sqrt{256+32r}\)
or, \(AD^{2}+DC^{2}=CA^{2}\)
Second Hint
or, \(48^{2}+(2r+16)^{2}\)
\(=(48+\sqrt{256+32r})^{2}\)
or, \(r^{2}+8r=24\sqrt{256+32r}\)
or, \(r(r+8)=24(4\sqrt{2})(\sqrt{r+8})\)
Final Step
or, r=24
or, 2r=diameter=48.
Try this beautiful problem from the Pre-RMO, 2019 based on Diameter of a circle.
A village has a circular wall around it, and the wall has four gates pointing north, southeast and west. A tree stands outside the village, 16 m north of the north gate, and it can be just seen appearing on the horizon from a point 48 m east of the south gate. Find the diameter in meters of the wall that surrounds the village.
Pythagoras Theorem
Equations
Integer
But try the problem first...
Answer: is 48.
PRMO, 2019, Question 25
Geometry Vol I to IV by Hall and Stevens
First hint
Let radius =r
or,\(AB=\sqrt{AO^{2}-OB^{2}}=\sqrt{(16+r)^{2}-r^{2}}\)
\(=\sqrt{256+32r}\)
or, \(AD^{2}+DC^{2}=CA^{2}\)
Second Hint
or, \(48^{2}+(2r+16)^{2}\)
\(=(48+\sqrt{256+32r})^{2}\)
or, \(r^{2}+8r=24\sqrt{256+32r}\)
or, \(r(r+8)=24(4\sqrt{2})(\sqrt{r+8})\)
Final Step
or, r=24
or, 2r=diameter=48.
