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The mosaic above was created in the fourteenth century for a wall in Morocco. Write an expression for each of the following parts of it shown in the detail below in terms of a or b or both a and b.
1. The perimeter of the square.
2. Its area.
3. The perimeter of the other rectangle.
4. Its area.
5. The perimeter of one of the trapezoids.
6. Its area.
The perimeter is just the sum of all the sides.
The area is length times height.
3. 4a + 4b.
Add all the sides of the shaded rectangle = a + b + a + b + a + b+ a + b = 4a + 4b.
4. a^2 + 2ab.
The length of the shaded rectangle = b + a + b = a + 2b.
The height of the shaded rectangle = a.
So, the area = a x (a + 2b) = a^2 + 2ab.
5. 4a + 2b
Traversing the trapezoid we get, a + a + a + b + a + b = 4a + 2b
6. ab+ b^2
Area of a trapezoid = (width 1 + width 2)/2 x height.
width 1 = a
width 2 = b + a + b = a + 2b
height = b.
So, area = (a + a + 2b)/2 x b = (2a + 2b)/2 x b = (a + b)b = ab+ b^2