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Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2011 based on Rectangles and sides.

In rectangle ABCD, AB=12 and BC=10 points E and F are inside rectangle ABCD so that BE=9 and DF=8, BE parallel to DF and EF parallel to AB and line BE intersects segment AD. The length EF can be expressed in theorem \(m n^\frac{1}{2}-p\) where m , n and p are positive integers and n is not divisible by the square of any prime, find m+n+p.

- is 107
- is 36
- is 840
- cannot be determined from the given information

Parallelograms

Rectangles

Side Length

But try the problem first...

Answer: is 36.

Source

Suggested Reading

AIME I, 2011, Question 2

Geometry Vol I to IV by Hall and Stevens

First hint

here extending lines BE and CD meet at point G and drawing altitude GH from point G by line BA extended till H GE=DF=8 GB=17

Second Hint

In a right triangle GHB, GH=10 GB=17 by Pythagorus thorem, HB=(\({{17}^{2}-{10}^{2}})^\frac{1}{2}\)=\(3({21})^\frac{1}{2}\)

Final Step

HA=EF=\(3({21})^\frac{1}{2}-12\) then 3+21+12=36.

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

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