Trapezium | Geometry | PRMO-2018 | Problem 5

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Geometry Problembased on Trapezium | PRMO-2018 | Problem 5
- Key Concepts
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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

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But try the problem first...

Answer:$84$

Source

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

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?