Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1999 based on Squares and triangles.

Squares and triangles – AIME I, 1999


The two squares share the same centre O and have sides of length 1, The length of AB is \(\frac{43}{99}\) and the area of octagon ABCDEFGH is \(\frac{m}{n}\) where m and n are relatively prime positive integers, find m+n.

Squares and Triangles
  • is 107
  • is 185
  • is 840
  • cannot be determined from the given information

Key Concepts


Squares

Triangles

Algebra

Check the Answer


But try the problem first…

Answer: is 185.

Source
Suggested Reading

AIME I, 1999, Question 4

Geometry Vol I to IV by Hall and Stevens

Try with Hints


First hint

Triangle AOB, triangleBOC, triangleCOD, triangleDOE, triangleEOF, triangleFOG, triangleGOH, triangleHOA are congruent triangles

Second Hint

with each area =\(\frac{\frac{43}{99} \times \frac{1}{2}}{2}\)

Final Step

then the area of all 8 of them is (8)\(\frac{\frac{43}{99} \times \frac{1}{2}}{2}\)=\(\frac{86}{99}\) then 86+99=185.

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