Try this beautiful problem from Geometry based on Trapezium.
Geometry Problem based on Trapezium | PRMO-2018 | Problem 5
Let ABCD be a trapezium in which AB||CD and AD is perpendicular on AB .suppose has an incircle which touches AB at Q and CD at P.Given that PC=36 and QB=49. Find PQ?
- $64$
- $81$
- $84$
Key Concepts
Geometry
Trapezoid
Circle
Check the Answer
But try the problem first…
Answer:$84$
PRMO-2018, Problem 5
Pre College Mathematics
Try with Hints
First hint
Let the radius of the inner circle be r
Therefore AQ=PD=r and AD=2r
Can you now finish the problem ……….
Second Hint
Draw a perpendicular from C on AB at the point F
Can you finish the problem……..
Final Step
Let the inner circle touches BC at E
Then CE=36 (as BE & BQ are tangents)
BE=49 (as CE & PC are tangents)
Let the radius of the inner circle be r
Therefore AQ=PD=r and AD=2r
Let draw a perpendicular from C on AB at the point F
So BF=(AB-AF)=(49-36)=13
BC=85
Now in the triangle CBF we have
\(CF^2=85^2-13^2\)
\(\Rightarrow CF=84\)
Therefore CF=PQ=84
Other useful links
- https://www.youtube.com/watch?v=dLHUPDQzc2Q
- https://www.cheenta.com/largest-and-smallest-numbers-amc-8-2006-problem-22/
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