Question

This figure is made up of a rectangle and parallelogram.

What is the area of this figure?



Enter your answer in the box. Do not round any side lengths.

Answer 1

The area of the figure is 40 units. (rounded off from 39.998 units)

Explanation:

1. Area of the figure = Area of Rectangle + Area of Parallelogram.

2. Naming points:

Let the rectangle consist of points A(-6,-1), B(-5,-5), C(3,-3) and D(2,1).

Let the parallelogram consist of points C(3,-3), D(2,1), E(2,7) and F(3,3).

3. Each coordinate has a point on the x-axis and a point on the y-axis. We shall refer to these points as x', x'' ,x''' and so on corresponding to the x-axis values of A, B, C and so on in that order till F. For the y-axis points, we will name them in the series of y', y'', y''' and so on.

In simple words, in A(-6,1), x' = -6; y' = -1

4. Formula for finding area:

Area of a rectangle = length (l) × width (w)

Area of a parallelogram = base (b) x height (h)

5. Finding area of the rectangle:

l = AD =
`\sqrt{[x'''' - x']^(2) - [y'''' - y']^(2)}`

, i.e. AD =
`\sqrt{[2 - (-6)]^(2) - [1 - (-1)]^(2)}`

, i.e. AD = √68 = 8.246 (rounded to 3 decimals) = l

Now, w = AB =
`\sqrt{[x'' - x']^(2) - [y'' - y']^(2)}`

, i.e. AB =
`\sqrt{[(-5) - (-6)]^(2) - [(-5) - (-1)]^(2)}`

, i.e. AB = √17 = 4.123 (rounded to 3 decimals) = w

Area of rectangle = l x w = 8.246 x 4.123 = 33.998 units.

6. Finding area of the parallelogram:

b = DE = 6 units (we can observe this from the graph, i.e. points from 1 to 7 on the y axis)

h = Height is the line drawn perpendicular to the base from point F, which as we can see is 1 unit long.

Hence, Area of parallelogram = b x h = 6 x 1 = 6 units.

7. Now, adding both, we can arrive at the final answer.

Area of the figure = Area of rectangle + Area of parallelogram,

i.e. 33.998 units + 6 units = 39.998 units. (can be rounded off to 40 units)