Question

The sides of this rectangle are 4x+8 and 2x+2 cm. (a)Find in terms of x cm. (i)the area of the whole rectangle. (ii) the area that is shaded area.(b)Show that the shaded and white regions have the same area.

Answer 1

The area of the whole rectangle and the shaded region is given in terms of
`\(x\)`. By algebraic manipulation, it is shown that the shaded and white regions have the same area, both being equal to zero.

**a) Finding the Areas in Terms of x:**

(i) The area of the whole rectangle is given by the product of its sides:

`\[ \text{Area of rectangle} = (4x + 8) * (2x + 2) \]`

(ii) To find the shaded area, we need to subtract the area of the white region from the area of the whole rectangle. The area of the white region is the product of its sides:

`\[ \text{Area of white region} = (2x + 2) * (4x + 8) \]`

Now, the shaded area is obtained by subtracting the area of the white region from the area of the whole rectangle:

`\[ \text{Shaded area} = (4x + 8) * (2x + 2) - (2x + 2) * (4x + 8) \]`

**b) Showing the Equality of Shaded and White Regions:**

To show that the shaded and white regions have the same area, we can simplify the expression for the shaded area:

`\[ \text{Shaded area} = (4x + 8) * (2x + 2) - (2x + 2) * (4x + 8) \]`

By factoring out a common factor of
`\((2x + 2)\)`, we get:

`\[ \text{Shaded area} = (2x + 2) * [(4x + 8) - (4x + 8)] \]`

Simplifying further, we have:

`\[ \text{Shaded area} = (2x + 2) * 0 = 0 \]`

This result shows that the shaded area is indeed equal to the white region's area, confirming that the two regions have the same area.