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Area of Equilateral Triangle | AIME I, 2015 | Question 4

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2015 from Geometry based on Area of Equilateral Triangle.

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2015 from Geometry, based on Area of Equilateral Triangle (Question 4).

Area of Triangle – AIME I, 2015


Point B lies on line segment AC with AB =16 and BC =4. Points D and E lie on the same side of line AC forming equilateral triangle ABD and traingle BCE. Let M be the midpoint of AE, and N be the midpoint of CD. The area of triangle BMN is x. Find \(x^{2}\).

Area of Triangle Problem
  • is 107
  • is 507
  • is 840
  • cannot be determined from the given information

Key Concepts


Algebra

Theory of Equations

Geometry

Check the Answer


Answer: is 507.

AIME, 2015, Question 4

Geometry Revisited by Coxeter

Try with Hints


Let A(0,0), B(16,0),C(20,0). let D and E be in first quadrant. then D =\((8,8\sqrt3)\), E=\((18,2\sqrt3\)).

M=\((9,\sqrt3)\), N=(\(14,4\sqrt3\)), where M and N are midpoints

since BM, BN, MN are all distance, BM=BN=MN=\(2\sqrt13\). Then, by area of equilateral triangle, x=\(13\sqrt3\) then\(x^{2}\)=507.

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