Try this beautiful problem from geometry based on Lengths of Rectangle Problem.

## Lengths of Rectangle Problem – AMC-10A, 2009- Problem 14

Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?

- \(3\)
- \(\sqrt 10\)
- \(2+\sqrt 2\)
- \(2\sqrt 3\)
- \(4\)

**Key Concepts**

Geometry

Square

Rectangle

## Check the Answer

But try the problem first…

Answer: \(3\)

AMC-10A (2009) Problem 14

Pre College Mathematics

## Try with Hints

First hint

Given that The area of the outer square is $4$ times that of the inner square.therefore we can say that Therefore the side of the outer square is $\sqrt 4 = 2$ times that of the inner square.Can you find out length of the longer side of each rectangle?

Can you now finish the problem ……….

Second Hint

Let the side length of the outer square is \(4x\) then the side length of the inner square be \(2x\).Hence the side length of the red region is \(2x\) .As the rectangles are congruent ,therefore side length of green shaded region and the side length of blue shaded region will be x

Therefore the length of the longer side of each rectangle be \(3x\) and length of the shoter side will be \(x\)

Final Step

Therefore the ratio of the length of the longer side of each rectangle to the length of its shorter side will be \(\frac{3x}{x}=3\)

## Other useful links

- https://www.cheenta.com/circular-cylinder-problem-amc-10a-2001-problem-21/
- https://www.youtube.com/watch?v=KgLep9tSYOU&t=1s

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