Ah, the wonders of geometry! Let's dive into the mysterious realm of dilations.
When a square of area 4 is dilated by a scale factor of k, we obtain a square of area 9. A dilation is essentially an enlargement or reduction of a shape while maintaining proportionality.
To find the sum of all possible values of k, let's start by finding the side length of the original square. The area of a square is given by the side length squared, so if the area is 4, that means the side length is 2.
When this square is dilated by a scale factor of k, the side length becomes 2k. The area of the dilated square is then (2k)^2 = 4k^2.
We're given that the area of the dilated square is 9, so we set up the equation:
4k^2 = 9
Now, let's solve for k. Taking the square root of both sides, we have:
2k = ±√9
2k = ±3
Dividing by 2, we find two possible values for k:
k = ±3/2
Now, let's sum up the possible values of k:
3/2 + (-3/2) = 0
So, the sum of all possible values of k is 0.
And there you have it! Dilations and square areas can be quite puzzling, but hey, at least we got to explore the world of geometry together. Keep those questions coming, and I'll be here to clown around with the answers!