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AIME I Geometry Math Olympiad USA Math Olympiad

Squares and Triangles | AIME I, 2008 | Question 2

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

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.

Squares and Triangles
  • 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.

Source
Suggested Reading

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.

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