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# Squares and Triangles | AIME I, 1999 | Question 4

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

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

But try the problem first…

Source

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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