When a square of area 4 is dilated by a scale factor of k, we obtain a square of area 9. Find the sum of all possible values of k.
I do not understand...it is not 3/2, as I was told, but I don't understand why? Help?
A square with an area of 4 has to have all sides equal to 2. Right?
If I increase by a factor of k that means multiply 2 times k
My area is 2k times 2k = 4k^2
I would set this equal to 9
I am getting 3/2 for the value of k
-3/2 is another possibility, but we have to discount it because we don't measure distance as negative numbers.
I agree with your 3/2, but I wonder if someone else summed -3/2 and 3/2 = 0
thanks a lot i appreciate your time
0 is the answer
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!
To find the sum of all possible values of k, we need to determine all the values of k that satisfy the given conditions.
The area of a square is found by squaring the length of one of its sides. Let's assume the original square has a side length of s.
According to the problem, the original square has an area of 4, so we have:
s^2 = 4
When dilated by a scale factor of k, the new square will have a side length of ks. The area of the new square is also given as 9, so we have:
(ks)^2 = 9
Expanding the equation, we get:
k^2 * s^2 = 9
Since we know that s^2 = 4 from the first equation, we can substitute this value into the second equation:
k^2 * 4 = 9
Dividing both sides of the equation by 4:
k^2 = 9/4
Now, to solve for k, we take the square root of both sides:
k = ± √(9/4)
Simplifying the square root, we have:
k = ±(3/2)
So the possible values of k are 3/2 and -3/2. The sum of all possible values of k is:
3/2 + (-3/2) = 0
Therefore, the sum of all possible values of k is 0.