Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2008 based on Squares and Triangles.

## Squares, triangles and Trapezium – AIME I, 2008

Square AIME has sides of length 10 units, isosceles triangle GEM has base EM, and the area common to triangle GEM and square AIME is 80 square units.Find the length of the altitude to EM in triangle GEM.

- is 107
- is 25
- is 840
- cannot be determined from the given information

**Key Concepts**

Squares

Trapezium

Triangles

## Check the Answer

But try the problem first…

Answer: is 25.

AIME I, 2008, Question 2

Geometry Revisited by Coxeter

## Try with Hints

First hint

let X and Y be points where the triangle intersects the square and [AXE]=[YIM]=\(\frac{100-80}{2}\)=10 then AX=YI=2 units then XY=10-4=6 units

Second Hint

triangle GXY is similar to triangle GEM where h=height of triangle GXY then by similarity \(\frac{h+10}{10}=\frac{h}{6}\)

Final Step

then h=15 and h+10=25.

## Other useful links

- https://www.cheenta.com/cubes-and-rectangles-math-olympiad-hanoi-2018/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

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