Determine the scale factor used in the dilation of line segment AB , centered at the origin.
A: (2,3)
B: (4,1)
To determine the scale factor of the dilation, we need to find the ratio of the lengths of the dilated line segment to the original line segment.
First, let's find the length of line segment AB. We can use the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
For A(2, 3) and B(4, 1):
dAB = √((4 - 2)^2 + (1 - 3)^2)
= √(2^2 + (-2)^2)
= √(4 + 4)
= √8 = 2√2
Next, let's find the length of the dilated line segment A'B'. Since the dilation is centered at the origin, the coordinates of A' and B' can be found by multiplying the coordinates of A and B by the scale factor. Let's assume the scale factor is k.
A' = (k * 2, k * 3) = (2k, 3k)
B' = (k * 4, k * 1) = (4k, k)
Using the distance formula again for A'(2k, 3k) and B'(4k, k):
dA'B' = √((4k - 2k)^2 + (k - 3k)^2)
= √(2k^2 + (-2k)^2)
= √(2k^2 + 4k^2)
= √(6k^2)
= √6 * k
Therefore, the scale factor used in the dilation of line segment AB, centered at the origin, is the ratio of the lengths:
Scale factor = dA'B' / dAB
= (√6 * k) / (2√2)
= (√6 / √2) * (k / 2)
= (√(6/2)) * (k / 2)
= √3 * (k / 2)
Thus, the scale factor used in the dilation of line segment AB, centered at the origin, is √3 * (k / 2).
What do you mean by √3 * (k / 2)?
I apologize for the confusion. In my previous response, I made an error in simplifying the expression. Let's correct that:
The correct scale factor for the dilation of line segment AB, centered at the origin, is √6/√2.
We can simplify this fraction by multiplying the numerator and denominator by √2:
√6/√2 * √2/√2 = (√6 * √2) / (√2 * √2) = √12 / 2
So, the scale factor used in the dilation of line segment AB, centered at the origin, is √12 / 2.
Please note that √12 can also be simplified further as 2√3. Therefore, the scale factor is also equal to 2√3 / 2, which simplifies to √3.
Therefore, the scale factor used in the dilation of line segment AB, centered at the origin, is √3.
It's wrong
I apologize for the mistake in my previous response. Let's correct it.
To determine the scale factor of the dilation, we need to find the ratio of the lengths of the dilated line segment to the original line segment.
First, let's find the length of line segment AB using the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
For A(2, 3) and B(4, 1):
dAB = √((4 - 2)^2 + (1 - 3)^2)
= √(2^2 + (-2)^2)
= √(4 + 4)
= √8 = 2√2
Next, let's find the length of the dilated line segment A'B'. Since the dilation is centered at the origin, the coordinates of A' and B' can be found by multiplying the coordinates of A and B by the scale factor. Let's assume the scale factor is k.
A' = (k * 2, k * 3) = (2k, 3k)
B' = (k * 4, k * 1) = (4k, k)
Using the distance formula again for A'(2k, 3k) and B'(4k, k):
dA'B' = √((4k - 2k)^2 + (k - 3k)^2)
= √(2k^2 + (-2k)^2)
= √(2k^2 + 4k^2)
= √(6k^2)
= √6 * k
Therefore, the scale factor used in the dilation of line segment AB, centered at the origin, is the ratio of the lengths:
Scale factor = dA'B' / dAB
= (√6 * k) / (2√2)
= (√6 / 2√2) * k
= (√6 / √(2 * 2)) * k
= (√6 / √4) * k
= (√6 / 2) * k
Thus, the scale factor used in the dilation of line segment AB, centered at the origin, is (√6 / 2) * k.
What is (√6 / 2) * k?
I apologize for the confusion in my previous responses. Let's simplify the expression (√6 / 2) * k correctly.
To simplify (√6 / 2) * k, we can rewrite it as (√6 * k) / 2. This means multiplying the square root of 6 and k, and then dividing the result by 2.
Thus, the scale factor used in the dilation of line segment AB, centered at the origin, is (√6 * k) / 2.
What is (√6 * k) / 2
To simplify the expression (√6 * k) / 2, we multiply the square root of 6 by k and then divide the result by 2.
This expression can be written as (√6 * k) / 2 = k√6 / 2.
Therefore, the simplified form of (√6 * k) / 2 is k√6 / 2.