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Algebra Arithmetic Geometry Math Olympiad PRMO

Area of a triangle | PRMO 2017 | Question 25

Try this beautiful problem from the Pre-RMO, 2017, Question 23, based on Solving Equation. You may use sequential hints to solve the problem.

Try this beautiful problem from the Pre-RMO, 2017, Question 25, based on Area of a triangle.

Area of triangles – PRMO 2017, Question 25


Let ABCD be a rectangle and let E and F be points on CD and BC respectively such that area (ADE) =16, area (CEF) =9, and area (ABF) =25. What is the area of triangle AEF?

  • is 107
  • is 30
  • is 840
  • cannot be determined from the given information

Key Concepts


Equation

Algebra

Integers

Check the Answer


But try the problem first…

Answer: is 30.

Source
Suggested Reading

PRMO, 2017, Question 25

Higher Algebra by Hall and Knight

Try with Hints


First hint

Let x=AD, a=DE, y-a=EC, b=FC, x-b=FB, y=AB

\(xa=32, b(y-a) =18, y(x-b) =50\)

or, \(by-\frac{32b}{x}=18\)

or, \(b=\frac{18b}{xy-32}\)

Area of a triangle problem

Second Hint

or, \(xy-\frac{18xy}{xy-32}=50\)

or, \(xy=t\)

or, \(\frac{t^{2}-32t-18t}{t-32}=50\)

Final Step

or, \(t^{2}-50t=50t-1600\)

or, \(t^{2}-100t+1600=0\)

or, t=80,20

or, xy=80

or, area of triangle AEF= 80-(16+9+25)=30.

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