Try this beautiful problem from Geometry: Ratio of the area between Square and Pentagon.

Ratio of the area between Square and Pentagon – AMC-8, 2013 – Problem 24


Squares ABCD ,EFGH and GHIJ are equal in area .Points C and D are the mid points of the side IH and HE ,respectively.what is the ratio of the area of the shaded pentagon AJICB to the sum of the areas of the three squares?

ratio of area of square and pentagon

  • $\frac{1}{4}$
  • $\frac{1}{3}$
  • $\frac{3}{8}$

Key Concepts


Geometry

Area of square

Area of Triangle

Check the Answer


But try the problem first…

Answer:$\frac{1}{3}$

Source
Suggested Reading

AMC-8(2013) Problem 24

Pre College Mathematics

Try with Hints


First hint

extend  IJ until it hits the extension of  AB .

Can you now finish the problem ……….

Second Hint

find the area of the pentagon

can you finish the problem……..

Final Step

solution figure

First let L=2 (where L is the side length of the squares) for simplicity. We can extend  IJ until it hits the extension of  AB . Call this point  X.

Then clearly length of AX=3 unit & length of XJ = 4 unit .

Therefore area of \(\triangle AXJ= (\frac{1}{2} \times AX \times XJ)=(\frac{1}{2} \times 4 \times 3)=6\) sq.unit

And area of Rectangle BXIC= \(( 1 \times 2)\)=2 sq.unit

Therefore the of the pentagon ABCIJ=6-2=4 sq.unit

The combined area of three given squares be \( (3 \times 2^2)\)=12 sq.unit

Now, the ratio of the shaded area(pentagon) to the combined area of the three squares is \(\frac{4}{12}=\frac{1}{3}\)

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