Find the area of the shaded region.
Let’s start this question by looking at the sheets in our diagram. It looks as though we have a kite internally and a parallelogram externally. But let’s be sure mathematically if this is the case. We can recall that the diagonals of a kite are perpendicular. And the longer diagonal bisects the shorter one. We can remember that perpendicular means at 90 degrees and to bisect means to cut exactly in half. Looking at our diagram, we can see that the diagonals do you have a 90-degree angle. And the two lines on our shorter diagonal indicate that these two lengths are the same size. This is, therefore, sufficient to say that we do indeed have a kite.
For a parallelogram, we would need to show that there are two pairs of opposite-sides parallel. In our diagram, we could see the annotations on the line indicating that our horizontal lines are parallel and our two other lines are also parallel, which is therefore sufficient to show that we have a parallelogram. So now, let’s work out how we would find the area of the shaded region. There are a number of ways in which we could find this area. But perhaps the simplest is to realize that the shaded region can be found by working out the area of the parallelogram and then subtracting the area of the kite.
To do this, we need to use two formulas. The first one that the area of a parallelogram is equal to the base times the height. And secondly, the area of a kite is equal to 𝑝 times 𝑞 over two where 𝑝 and 𝑞 are the diagonals of the kite. So let’s start then with our parallelogram. We can see that our base is 30 feet and our height is 25 feet. So 30 times 25 is 750 square feet. And next, to find the area of the kite, we multiply the two diagonals and divide by two, which is 30 times 25 over two giving us an answer of 375 square feet. We can now use these values to find the area of the shaded region. Therefore, the area of the shaded region is 750 take away 375 giving us a final answer of 375 square feet.
We may notice at this point that the shaded area is half of the area of the parallelogram, although this may not be particularly clear in the diagram. So let’s see if we can verify this answer. If we take the area of a parallelogram to be the base times the height, then we could also see the area of a rectangle with the same base and height would be equivalent. This could give us a clearer picture of the fact that the area of the kite will be half that off the rectangle or parallelogram in which it sits. And so we have official confirmation of our answer of 375 square feet.