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.

- 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.

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.

## Other useful links

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

Google