ABCD is a square with sides of 20 centimeters. AX=XB, BY=YC, CZ=ZD, AW=WD and XZ are straight lines. Find the total area of the shaded parts
Let's call the shaded parts A1, A2, A3, and A4.
A1:
The area of A1 is equal to the area of triangle AXD. Since AX = XD and AD = 20 cm (the side length of the square), triangle AXD is an isosceles right triangle. The area of a right triangle is given by (1/2) * base * height, so the area of triangle AXD is (1/2) * 20 cm * 20 cm = 200 cm^2.
A2:
The area of A2 is equal to the area of the square ABCD minus the area of the triangle AXD. The area of the square is 20 cm * 20 cm = 400 cm^2, and we already know the area of triangle AXD is 200 cm^2, so the area of A2 is 400 cm^2 - 200 cm^2 = 200 cm^2.
A3:
The area of A3 is equal to the area of triangle AWZ. Since AW = WD = 20 cm (the side length of the square), and AZ is a straight line connecting the midpoints of AW and WD, triangle AWZ is a right triangle with legs of length 20 cm. The area of a right triangle is given by (1/2) * base * height, so the area of triangle AWZ is (1/2) * 20 cm * 20 cm = 200 cm^2.
A4:
The area of A4 is equal to the area of the square ABCD minus the area of triangle AWZ. The area of the square is 20 cm * 20 cm = 400 cm^2, and we already know the area of triangle AWZ is 200 cm^2, so the area of A4 is 400 cm^2 - 200 cm^2 = 200 cm^2.
To find the total area of the shaded parts, we add up the areas of A1, A2, A3, and A4:
Total area = A1 + A2 + A3 + A4
Total area = 200 cm^2 + 200 cm^2 + 200 cm^2 + 200 cm^2
Total area = 800 cm^2
Therefore, the total area of the shaded parts is 800 cm^2.
To find the total area of the shaded parts in the square ABCD, we need to partition the square and calculate the areas of each shaded region separately.
Step 1: Draw the square ABCD and label the points as given in the question: A, B, C, and D. Draw the lines AX, BX, CY, and DZ.
Step 2: Divide the square into smaller regions as follows:
- Region 1 (shaded in orange): This is the region outside the smaller square in the center. It consists of 4 identical right-angled triangles. The area of one of these triangles can be calculated as (1/2) * 20 * 20 = 200 square centimeters. Therefore, the total area of Region 1 is 4 * 200 = 800 square centimeters.
Step 3: Find the area of the smaller square in the center (shaded in blue). Each side of this square is equal to half the length of the side of the larger square, so the side length is 20 / 2 = 10 centimeters. Therefore, the area of this square is 10 * 10 = 100 square centimeters.
Step 4: Find the area of Region 2 (shaded in green). This region consists of two identical right-angled triangles. The base of each triangle is 10 centimeters (the side length of the smaller square), and the height is the same as the side length of the larger square, which is 20 centimeters. Therefore, the area of each triangle is (1/2) * 10 * 20 = 100 square centimeters. The total area of Region 2 is 2 * 100 = 200 square centimeters.
Step 5: Find the area of Region 3 (shaded in yellow). This region consists of two identical right-angled triangles. The base of each triangle is the same as the side length of the larger square, which is 20 centimeters, and the height is also 20 centimeters. Therefore, the area of each triangle is (1/2) * 20 * 20 = 200 square centimeters. The total area of Region 3 is 2 * 200 = 400 square centimeters.
Step 6: Add the areas of all the shaded regions to find the total shaded area: 800 + 100 + 200 + 400 = 1,500 square centimeters.
So, the total area of the shaded parts in the square ABCD is 1,500 square centimeters.