This was originally posted on another student's thread.
Express the side length of a square as a function of the length d of the square’s diagonal. Then express the area as a function of the diagonal length
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posted by Anonymous
today at 3:17pm
yA mUm gEiy ;))
side length = s
then area = s^2
but
d^2 = s^2 + s^2 = 2 s^2
so
s^2 = area = d^2/2
To express the side length of a square as a function of the length d of the square's diagonal, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (diagonal) is equal to the sum of the squares of the lengths of the other two sides. In the case of the square, the two other sides are the sides of the square, which are equal in length.
Let's assume the length of one side of the square is represented by 's'. According to the Pythagorean theorem, we can express the length d of the diagonal as:
d^2 = s^2 + s^2
Simplifying this equation, we get:
d^2 = 2s^2
To express the side length 's' (also known as the Square Root of Square Root of d^2 divided by 2, or \sqrt[4]{d^2/2}).
To express the area of the square as a function of the diagonal length 'd', we use the formula:
Area = s^2
Substituting the expression for side length 's' we derived earlier, we get:
Area = (\sqrt[4]{d^2/2})^2
Simplifying this equation, we get:
Area = (d^2/2)^{1/2}
Therefore, the area of the square can be expressed as the square root of one-half of the square of the length of the diagonal.