Geometry of                      and1. Consider a square whose side is a + b.2. It is composed of                 a square...
3. Notice the shaded areas in the diagram. The gray and hatched rectangles are both ab.        They overlap each other, an...
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Geometry of (a b) squared

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Geometry of (a b) squared

  1. 1. Geometry of and1. Consider a square whose side is a + b.2. It is composed of a square whose side is a, a square whose side is b, and two rectangles ab.3. That is, (a + b)² = a² + 2ab + b².How to teach ? 1. First draw a square with sides a, which is . 2. Draw lines to cut the sides of the square into two lengths: b and the remainder, . This defines the square .
  2. 2. 3. Notice the shaded areas in the diagram. The gray and hatched rectangles are both ab. They overlap each other, and the area of the overlap is . 4. Therefore, to make the square , start with . Cut off and throw away the rectangle ab. Now, you want to throw away another rectangle ab. But to make the second ab, you will first have to add in an extra . 5. This leaves only our goal, the square , which shows geometrically that: .Explanation 1. The Figure shows the big square with the side length a and the two congruent "long" rectangles with the side sizes a and b each (they include the square with the side b). 2. The areas of the two smaller squares are and respectively, while the area of the large square is . 3. The area of each of the two "long" rectangles is . 4. Now, if we distract the areas of the two "long" rectangles from the area of the big square , the area of the small square will be distracted two times. So, let us add the area one time to compensate it. 5. In this way, we will get exactly the area of the square . This is exactly what the formula for the square of a difference says: .

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