## Outline Mathematics

Geometry

# Three Congruent Rectangles

A geometry problem that requires a little logic and algebra:

Reactangle ABCD contains three small congruent rectangles. If the smaller dimension of one of the small rectangles is 5 units, what is the area of rectangle ABCD in square units? |

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Copyright © 1996-2018 Alexander Bogomolny

### Solution

Reactangle ABCD contains three small congruent rectangles. If the smaller dimension of one of the small rectangles is 5 units, what is the area of rectangle ABCD in square units? |

We may make three observations:

- Sides AD and BC,AD,AB,BC,CD of the big rectangle are equal.
- BC also serves as the big,big,small side of the small rectangle.
- (Looking at AD,) Two small sides of a small rectangle fit exactly in the small,big,small side of the big rectangle.

From #3, the small side of the big rectangle is twice,equal,twice,thrice the small side of the small rectangle, i.e., 10,5,10,20,30 units. From #2 (and #1), the big side of the small rectangle equals 10 units. The area of a rectangle equals the product of its sides. Therefore, the area of a small rectangle equals

Let's do this in a little more general way. Let x and y denote the small and the large dimensions of the small rectangle. This makes the area of the small rectangle xy,x + y,2x + y,3xy,xy and the area of the big rectangle 3xy. On the other hand, the small side of the big rectangle is 2x,x,2x,3x,4x whereas its big side measures

3xy = 2x·(x + y). |

If x = 0, the problem degenerates into a case that requires no calculations. The big rectangles fills zero area as does a small rectangle. So assume

3y = 2(x + y) = 2x + 2y, |

which shows that

y = 2x. |

For x = 5, y = 10,5,10,20,30 and we are done.

### References

- G. Lenchner,
*Math Olympiad Contest Problems For Elementary and Middle Schools*, Glenwood Publications, NY, 1997

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Copyright © 1996-2018 Alexander Bogomolny