cuts across on diagnal saves 1/3 distance of longer side what ratio of short to long side
To find the ratio of the short side to the long side, we need to first determine the length of the cut across the diagonal of the shape.
Assuming we have a rectangle as the shape in question, we can use the Pythagorean theorem to find the length of the diagonal. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Let's assume that the shorter side of the rectangle is represented by 'a', and the longer side is represented by 'b'. The diagonal cuts across the rectangle, forming a right triangle with the two sides of the rectangle as its legs.
Applying the Pythagorean theorem, we have:
a^2 + b^2 = diagonal^2
Since the diagonal saves 1/3 distance of the longer side, the length of the diagonal is (2/3)b.
Substituting this value into the equation, we get:
a^2 + b^2 = (2/3)b^2
Now, we can solve this equation to find the relationship between the short and long sides.
Rearranging the equation, we have:
a^2 = (2/3)b^2 - b^2
a^2 = (2/3 - 1)b^2
a^2 = (-1/3)b^2
To eliminate the square root, we can divide both sides of the equation by b^2:
a^2 / b^2 = -1/3
Taking the square root of both sides, we obtain:
|a/b| = sqrt(-1/3)
At this point, we need to consider the square root of a negative number. Since the square root of -1 is denoted by 'i' (imaginary unit), the square root of -1/3 can be denoted as i * sqrt(1/3).
So, the ratio of the short side (a) to the long side (b) of the rectangle is:
|a/b| = i * sqrt(1/3)
Note that the absolute value signs are used around a/b because the length of a side cannot be negative.