Three Rectangles

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? So I have five three-dimensional shapes over here and I also have five names for them and what I want you to do is pause this video and think about which of these shapes is a square pyramid which of these is a rectangular prism which one is a triangular prism which one is a sphere and which one is a cylinder all right now let's just work through this together and really this is just something.

1. 3 Shape Phone Number
2. Peg Games Involving Three Rectangles
3. All 3 Dimensional Shapes

Outline Mathematics
Geometry

1. From 0 to 3 by using three right rectangles. See the figure below. The heights of the three rectangles are given by the function values at their right edges: f (1) = 2, f (2) = 5, and f (3) = 10. Each rectangle has a width of 1, so the areas are 2, 5, and 10, which total 17.
2. A root rectangle is a rectangle in which the ratio of the longer side to the shorter is the square root of an integer, such as √ 2, √ 3, etc. The root-2 rectangle (ACDK in Fig. 10) is constructed by extending two opposite sides of a square to the length of the square's diagonal.
3. A prism is a three-dimensional figure with two parallel, congruent bases. The bases, which are also two of the faces, can be any polygon. The other faces are rectangles. A prism is named according to the shape of its bases. A pyramid is a three-dimensional figure with only one base. The base can be any polygon.

A geometry problem that requires a little logic and algebra:

Copyright © 1996-2018 Alexander Bogomolny

Solution

We may make three observations:

1. Sides AD and BC,AD,AB,BC,CD of the big rectangle are equal.
2. BC also serves as the big,big,small side of the small rectangle.
3. (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 5,5,10,20,30·10 = 50,50,60,70,80 square units.Three,Two,Three,Four small rectangles fit into the big one, making its area three times as large. It follows that the area of rectangle ABCD equals 3·50 = 150 unit².

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 x + y,x + y,2x + y,3xy,xy. It follows that the area of rectangle ABCD is also given by 2x·(x + y). The two quantities are equal:

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 x ≠ 0,0,1,2. This assumption allows us to divide both sides of the equation by x:

which shows that

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

References

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

Copyright © 1996-2018 Alexander Bogomolny

68344339

(Jump to Area of a Rectangle or Perimeter of a Rectangle)

A rectangle is a four-sided flat shape where every angle is a right angle (90°).

Play with a rectangle:

1. From 0 to 3 by using three right rectangles. See the figure below. The heights of the three rectangles are given by the function values at their right edges: f (1) = 2, f (2) = 5, and f (3) = 10. Each rectangle has a width of 1, so the areas are 2, 5, and 10, which total 17.
2. A root rectangle is a rectangle in which the ratio of the longer side to the shorter is the square root of an integer, such as √ 2, √ 3, etc. The root-2 rectangle (ACDK in Fig. 10) is constructed by extending two opposite sides of a square to the length of the square's diagonal.
3. A prism is a three-dimensional figure with two parallel, congruent bases. The bases, which are also two of the faces, can be any polygon. The other faces are rectangles. A prism is named according to the shape of its bases. A pyramid is a three-dimensional figure with only one base. The base can be any polygon.

A geometry problem that requires a little logic and algebra:

Copyright © 1996-2018 Alexander Bogomolny

Solution

We may make three observations:

1. Sides AD and BC,AD,AB,BC,CD of the big rectangle are equal.
2. BC also serves as the big,big,small side of the small rectangle.
3. (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 5,5,10,20,30·10 = 50,50,60,70,80 square units.Three,Two,Three,Four small rectangles fit into the big one, making its area three times as large. It follows that the area of rectangle ABCD equals 3·50 = 150 unit².

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 x + y,x + y,2x + y,3xy,xy. It follows that the area of rectangle ABCD is also given by 2x·(x + y). The two quantities are equal:

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 x ≠ 0,0,1,2. This assumption allows us to divide both sides of the equation by x:

which shows that

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

References

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

Copyright © 1996-2018 Alexander Bogomolny

68344339

(Jump to Area of a Rectangle or Perimeter of a Rectangle)

A rectangle is a four-sided flat shape where every angle is a right angle (90°).

Play with a rectangle:

Area of a Rectangle

Example: A rectangle is 6 m wide and 3 m high, what is its Area?

Perimeter of a Rectangle

The Perimeter is the distance around the edges.

Example: A rectangle is 12 cm long and 5 cm tall, what is its Perimeter?

Diagonals of a Rectangle

3 Shape Phone Number

Example: A rectangle is 12 cm wide, and 5 cm tall, what is the length of a diagonal?

Peg Games Involving Three Rectangles

Golden Rectangle

All 3 Dimensional Shapes

There is also a special rectangle called the Golden Rectangle: