M and N are the midpoints of the sides of a square. What is the ratio of the area of triangle AMN to the area of the complete square?
Area of a triangle = ( 1 / 2 ) base * heigh
In this case base = heigh = a / 2
A = ( 1 / 2 ) * ( a / 2 ) * ( a / 2 )
A = a ^ 2 / 8
Area of a square = a ^ 2
( a ^ 2 / 8 ) / a ^ 2 = 1 / 8
To find the ratio of the area of triangle AMN to the area of the complete square, we can use the following steps:
1. First, let's label the length of one side of the square as 's'. Since M and N are the midpoints of the sides, the length of AM and AN is s/2.
2. The area of triangle AMN can be found using the formula: A = (1/2) * base * height. In this case, the base is AM and the height is AN.
So, the area of triangle AMN is: A_AMN = (1/2) * (s/2) * (s/2) = (s^2)/8.
3. The area of the complete square is given by: A_square = s^2.
4. Finally, the ratio of the area of triangle AMN to the area of the complete square can be calculated as:
Ratio = A_AMN / A_square = ((s^2)/8) / (s^2) = (s^2)/(8 * s^2) = 1/8.
Therefore, the ratio of the area of triangle AMN to the area of the complete square is 1/8.
To find the ratio of the area of triangle AMN to the area of the complete square, we need to calculate the area of triangle AMN as well as the area of the square.
Let's start by calculating the area of the square. Since we don't know any specific measurements of the square, we can assume that the length of one side is "s". In that case, the area of the square would be s^2.
Now, let's consider triangle AMN. Since M and N are the midpoints of the sides of the square, triangle AMN is half of one of the four smaller triangles formed by dividing the square diagonally.
To calculate the area of triangle AMN, we need to know the length of one of its sides. Since M and N are midpoints, they divide each side of the square into halves. So the length of one side of triangle AMN is s/2.
Now, we can calculate the area of triangle AMN using the formula: Area = (base * height) / 2.
The base of triangle AMN is s/2 and the height is s/2. Substituting these values into the formula, we get:
Area of triangle AMN = (s/2 * s/2) / 2 = s^2/8
Finally, to find the ratio of the area of triangle AMN to the area of the complete square, we divide the area of triangle AMN by the area of the square:
Ratio = (s^2/8) / (s^2) = 1/8
Therefore, the ratio of the area of triangle AMN to the area of the complete square is 1/8.