In the figure, 𝑋𝑌𝑍𝑊 is a rhombus and 𝐴𝐵𝐶𝐷 a rectangle. Find the shaded area, given that 𝑋𝑍 equals four, 𝑌𝑊 equals six, and 𝐴𝐵 equals 10.
We can start by labeling what we know. 𝐴𝐵 equals 10, 𝑋𝑍 equals four, and 𝑌𝑊 equals six. If we’re interested in the shaded area, we’ll need to find the area of the rectangle and then take away the area of the rhombus. The shaded area is equal to the area of the rectangle with the part that is the rhombus taken out.
To go about this, we’ll need to know how to find the area of the rectangle, which is found by multiplying its length times its width. And then by finding the area of a rhombus, which is equal to 𝑝 times 𝑞 over two, where 𝑝 and 𝑞 are the diagonals of the rhombus.
If we start with our rectangle, we know that 𝐴𝐵 equals 10. And so the length equals 10. But what about 𝐴𝐷 or 𝐵𝐶? What about the width? Because 𝑋 and 𝑍 fall on opposite sides of the rectangle, we can say that the width of this rectangle will be four at all places. 𝐴𝐷 will equal four, and 𝐵𝐶 will equal four. And that means, in finding the area, we would multiply 10 times four. The width is four. The area of our rectangle would be 40 units squared.
We should follow the same process for the rhombus. We know that six and four are the lengths of the diagonals of this rhombus. And so the area of the rhombus will be six times four divided by two, which would be 12 units squared. Now we plug in what we know. The area of the rectangle is 40. The area of the rhombus is 12. 40 minus 12 will tell us the area of the shaded region, which is 28.