Try this beautiful problem from the Pre-RMO, 2017, Question 25, based on Area of a triangle.
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?
Equation
Algebra
Integers
But try the problem first...
Answer: is 30.
PRMO, 2017, Question 25
Higher Algebra by Hall and Knight
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, \(b=\frac{18b}{xy-32}\)
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.
Try this beautiful problem from the Pre-RMO, 2017, Question 25, based on Area of a triangle.
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?
Equation
Algebra
Integers
But try the problem first...
Answer: is 30.
PRMO, 2017, Question 25
Higher Algebra by Hall and Knight
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, \(b=\frac{18b}{xy-32}\)
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.
