The endpoints of are A(2, 2) and B(3, 8). is dilated by a scale factor of 3.5 with the origin as the center of dilation to give image . What are the slope (m) and length of ?

To find the slope (m) and length of the image CD, we need to follow these steps:

Step 1: Find the slope of the line AB using the formula:
m = (y₂ - y₁) / (x₂ - x₁)

Given points A(2, 2) and B(3, 8), we can substitute the values into the formula:
m = (8 - 2) / (3 - 2)
m = 6 / 1
m = 6

So the slope of line AB is 6.

Step 2: Since the origin is the center of dilation, the coordinates of point A and B do not change. To find the coordinates of point C, we need to perform the dilation.
For point C:
x-coordinate of C = x-coordinate of A * scale factor = 2 * 3.5 = 7
y-coordinate of C = y-coordinate of A * scale factor = 2 * 3.5 = 7

For point D:
x-coordinate of D = x-coordinate of B * scale factor = 3 * 3.5 = 10.5
y-coordinate of D = y-coordinate of B * scale factor = 8 * 3.5 = 28

So, the coordinates of C are (7, 7) and the coordinates of D are (10.5, 28).

Step 3: Calculate the length of line CD using the distance formula:
d = sqrt((x₂ - x₁)² + (y₂ - y₁)²)

Substituting the values, we have:
d = sqrt((10.5 - 7)² + (28 - 7)²)
d = sqrt(3.5² + 21²)
d = sqrt(12.25 + 441)
d = sqrt(453.25)
d ≈ 21.3

Therefore, the slope of line CD is 6 and the length of line CD is approximately 21.3.

To find the slope and length of the image of the line segment, we'll first find the image of the endpoints under the dilation.

Step 1: Find the image of point A.
The image of point A under dilation is obtained by multiplying the coordinates of A by the scale factor of 3.5. Since the origin is the center of dilation, the image of A will have coordinates (3.5 * 2, 3.5 * 2). Therefore, the image of A is A'(7, 7).

Step 2: Find the image of point B.
Similar to point A, the image of point B under dilation is obtained by multiplying the coordinates of B by 3.5. So the image of B will have coordinates (3.5 * 3, 3.5 * 8), which simplifies to B'(10.5, 28).

Step 3: Determine the slope of the line segment AB.
The slope of a line can be calculated using the formula:
slope (m) = (change in y-coordinate) / (change in x-coordinate)

To find the change in y-coordinate, we subtract the y-coordinate of A' from the y-coordinate of B':
change in y-coordinate = 28 - 7 = 21

Similarly, we find the change in x-coordinate by subtracting the x-coordinate of A' from the x-coordinate of B':
change in x-coordinate = 10.5 - 7 = 3.5

Therefore, the slope of the line segment AB' is:
m = change in y-coordinate / change in x-coordinate = 21 / 3.5 = 6

Step 4: Calculate the length of the line segment AB'.
The length of a line segment can be calculated using the distance formula:
Length = square root [ (x2 - x1)^2 + (y2 - y1)^2 ]

Plugging in the coordinates of A' (7, 7) and B' (10.5, 28), we get:
Length = square root [ (10.5 - 7)^2 + (28 - 7)^2 ]
= square root [ 3.5^2 + 21^2 ]
= square root [ 12.25 + 441 ]
= square root [ 453.25 ]
≈ 21.29

Therefore, the slope (m) of the line segment AB' is 6, and the length of the line segment AB' is approximately 21.29.

The slope of a line does not change.

The length expands by the dilation factor.