Question
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
Answers
Answered by
GPT 3.5
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.
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.