Point A (4, 8) is mapped to point D (10, 20) with a dilation centered at the origin. Which of the following is the scale for this dilation?
note that both coordinates are multiplied by 2.5
To find the scale for the dilation centered at the origin, we can use the formula:
scale = distance from origin to new point / distance from origin to original point
Let's calculate the distances:
Distance from origin to point A:
OA = sqrt(x₁² + y₁²)
= sqrt(4² + 8²)
= sqrt(16 + 64)
= sqrt(80)
= 4√5
Distance from origin to point D:
OD = sqrt(x₂² + y₂²)
= sqrt(10² + 20²)
= sqrt(100 + 400)
= sqrt(500)
= 10√5
Now, we can calculate the scale:
scale = OD / OA
= (10√5) / (4√5)
= 10/4
= 5/2
Therefore, the scale for this dilation is 5/2.
To determine the scale of the dilation, we need to consider the relationship between the coordinates of the original point A (4, 8) and the corresponding coordinates of the dilated point D (10, 20).
The scale of a dilation is found by taking the ratio of the corresponding side lengths or distances from the center of dilation. In this case, since the dilation is centered at the origin (0, 0), we can determine the scale by comparing the lengths of the corresponding sides centered at the origin.
The x-coordinate ratio of the dilation can be found by dividing the x-coordinate of D (10) by the x-coordinate of A (4):
x-coordinate ratio = (x-coordinate of D) / (x-coordinate of A) = 10 / 4 = 2.5
Similarly, the y-coordinate ratio of the dilation can be found by dividing the y-coordinate of D (20) by the y-coordinate of A (8):
y-coordinate ratio = (y-coordinate of D) / (y-coordinate of A) = 20 / 8 = 2.5
Since both the x-coordinate and y-coordinate ratios are equal to 2.5, we can conclude that the scale of this dilation is 2.5.