If a square is dilated from a point inside by a scale factor of 3, how does the length of the diagonal of the original square compare to the length of the diagonal of the dilated square?
To find the relationship between the length of the diagonal of the original square and the length of the diagonal of the dilated square, we need to understand how dilation affects the dimensions of a figure.
Dilation is a transformation that enlarges or reduces the size of a figure by a certain scale factor from a fixed center point. In this case, the center of dilation is inside the square.
Let's start by determining the lengths of the diagonals. In a square, the length of the diagonal can be found using the Pythagorean theorem.
The Pythagorean theorem states that for a right triangle, the square of the length of the hypotenuse (the diagonal) is equal to the sum of the squares of the lengths of the other two sides.
In a square, the diagonal forms a right triangle with two sides equal to the length of one side of the square. Let's call the length of one side of the square "s."
Using the Pythagorean theorem, we can find the length of the diagonal of the original square. The two sides of the right triangle are equal to "s," so:
(diagonal of the original square)^2 = s^2 + s^2
(diagonal of the original square)^2 = 2s^2
Taking the square root of both sides, we get:
diagonal of the original square = √(2s^2)
diagonal of the original square = √2 * s
Now, let's consider the dilated square with a scale factor of 3. The sides of the dilated square will be three times longer than the sides of the original square.
Let "3s" represent the length of one side of the dilated square.
Using the same logic as before, we can find the length of the diagonal of the dilated square:
(diagonal of the dilated square)^2 = (3s)^2 + (3s)^2
(diagonal of the dilated square)^2 = 9s^2 + 9s^2
(diagonal of the dilated square)^2 = 18s^2
Again, taking the square root of both sides, we get:
diagonal of the dilated square = √(18s^2)
diagonal of the dilated square = √18 * s
Comparing the two diagonals, we have:
(diagonal of the dilated square) / (diagonal of the original square) = (√18 * s) / (√2 * s)
(diagonal of the dilated square) / (diagonal of the original square) = (√(18/2)) * (s/s)
(diagonal of the dilated square) / (diagonal of the original square) = √9
(diagonal of the dilated square) / (diagonal of the original square) = 3
Therefore, the length of the diagonal of the dilated square is three times the length of the diagonal of the original square.