Question 15. The diagram shows two large squares with the same area. Part of each square is shaded, as shown. In the first square, the midpoints of the adjacent sides are joined. In the second square, four smaller squares, all with side lengths equal to a third of the side length of the large square, are shaded. The area shaded in the first square is 9. What is the area shaded in the second square? First, we can draw two lines in the first square and use them, along with the existing lines, to divide the square into eight equal triangles. Notice how four of these triangles, or exactly half of them, are shaded. Since half of the triangles are shaded, the total area of the square must be double of the shaded region. Multiplying 9, the area of the shaded region, by 2, gives us 18 for the area of the square. Now, let's look at the second square. We are told that these smaller shaded squares have exactly one third the side length of the larger square. This means that we can draw these four lines to divide the square into nine equal smaller squares. Since the large square is the same area as the first square, we know that the area of these nine smaller squares must equal 18. This means that each individual small square has an area of 18 divided by 9, which is equal to 2. Since four of these smaller squares are shaded, we can multiply 2 by 4 to get the total shaded area, which is 8.

geometry

square

shaded area

area calculation

mathematics