A big square is divided into 7 identical square and 2 identical triangles . The are of the shaded small square is 4cm square, what is the total area of the shaded parts in cm square ?
To find the total area of the shaded parts, we need to determine the area of each individual shaded part and then add them together.
First, let's find the area of one of the identical squares. Since the big square is divided into 7 identical squares, the area of each square will be 1/7th of the total area of the big square. Let's call the area of each square "A".
Since the big square is a square, all of its sides are equal. Let's call the side length of the big square "s".
Therefore, the total area of the big square is s^2, and the area of each of the identical small squares is (1/7)*s^2.
Now let's look at the two identical triangles. Since they are identical, the area of each triangle will be the same. Let's call the area of each triangle "T".
We know that the total area of the shaded small square is 4 cm^2. Let's call the side length of the shaded small square "x". Therefore, the area of the shaded small square is x^2.
Now, let's set up an equation using the information we have:
(1/7)*s^2 + (1/7)*s^2 + 2T + x^2 = 4 cm^2
Simplifying the equation, we have:
(2/7)*s^2 + 2T + x^2 = 4 cm^2
Since each small square and triangle is identical, we can rewrite the equation as:
(2/7)*A + 2T + 4 = 4 cm^2
Subtracting 4 from both sides, we have:
(2/7)*A + 2T = 0 cm^2
Since the area of the shaded small square is 4 cm^2, the equation becomes:
(2/7)*A + 2T = 4 cm^2
Now, let's find the area of one of the small squares using the equation.
Let's say that the area of each small square is A. Since the big square is divided into 7 identical squares, the area of each small square is (1/7)*A.
Let's plug this value into the equation:
(2/7)*(1/7)*A + 2T = 4 cm^2
Simplifying, we have:
(2/49)*A + 2T = 4 cm^2
Since A is cancelled out, the equation becomes:
2T = 4 cm^2
Dividing both sides by 2, we find:
T = 2 cm^2
So, the area of each triangle is 2 cm^2.
Now, let's find the area of one of the small squares.
Using the equation:
(1/7)*s^2 = 4 cm^2
Multiplying both sides by 7, we have:
s^2 = 28 cm^2
Taking the square root of both sides, we find:
s = √28 cm
Therefore, the side length of each small square is √28 cm.
Now, we know the area of each small square is (1/7)*s^2 and the area of each triangle is T.
The total area of the shaded parts is given by:
(2/7)*A + 2T
Substituting the values we found, we have:
(2/7)*(1/7)*s^2 + 2T = (2/7)*(1/7)*28 + 2*2 = 8 + 4 = 12 cm^2
Therefore, the total area of the shaded parts is 12 cm^2.