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# Area of a triangle | PRMO 2017 | Question 25

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

PRMO, 2017, Question 25

Higher Algebra by Hall and Knight

## Try with Hints

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, $b=\frac{18b}{xy-32}$

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

or, $xy=t$

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

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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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

PRMO, 2017, Question 25

Higher Algebra by Hall and Knight

## Try with Hints

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, $b=\frac{18b}{xy-32}$

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

or, $xy=t$

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

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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