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? But the answer has to be in a ratio which is the area of a triangle to the area of the complete square. Also the triangle is in the square at top corner.
Let x = length of side of square
thus area of square is
A,square = x^2
recall that area of triangle is given by
A,triangle = (1/2)b*h
where b is the base and h is the height.
The base and height of the triangle AMN are equal, which is (1/2)x. Thus,
A,triangle = (1/2)*[(1/2)x]*[(1/2)x]
A,triangle = (1/8)x^2
Getting the ratio of areas,
A,triangle : A,square = (1/8)x^2 : x^2 = 1/8
hope this helps~ :)
To find the ratio of the area of triangle AMN to the area of the complete square, we first need to determine the relationship between the triangle and the square.
Since M and N are the midpoints of the sides of the square, we can draw two diagonals within the square that intersect at its center. These diagonals divide the square into four congruent right triangles.
Now, let's consider triangle AMN in relation to one of these right triangles.
Since M is the midpoint of one side of the square, triangle AMN has a base equal to half the length of the side of the square. Let's call this side length "s."
The height of triangle AMN is the distance between point A (the top corner of the square) and the line segment MN, which is the median of the right triangle. This distance is equal to half the length of the diagonal of the square, which we can call "d."
Thus, the area of triangle AMN can be calculated as:
Area of triangle AMN = (1/2) * base * height
= (1/2) * (s/2) * d
= s * d / 4
Now, let's find the area of the complete square.
The area of the square is given by the formula:
Area of square = side * side
= s * s
To get 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 * d / 4) / (s * s)
= d / (4 * s)
Therefore, the ratio of the area of triangle AMN to the area of the complete square is "d / (4 * s)."