Question

line segment AB coordinates are: A(2,4) B(6,6)

A dilation of line segment AB occurs using a scale factor of 1/2 and a center of dilation at the origin. Prove that Line segment A'B' is half the length of line segment AB by finding the points of A' and B'

Answer 1

So what are the coordinates of line segment A' B'?

Answer 2

To find the points \( A' \) and \( B' \) after the dilation of line segment \( AB \) with coordinates \( A(2,4) \) and \( B(6,6) \), using a scale factor of \( \frac{1}{2} \) from the center of dilation at the origin, we can use the following formula for dilation:

\[
(x', y') = (k \cdot x, k \cdot y)
\]

where \( k \) is the scale factor. Here, \( k = \frac{1}{2} \).

### Finding Coordinates of \( A' \):

For point \( A(2, 4) \):

\[
A' = \left( \frac{1}{2} \cdot 2, \frac{1}{2} \cdot 4 \right) = (1, 2)
\]

### Finding Coordinates of \( B' \):

For point \( B(6, 6) \):

\[
B' = \left( \frac{1}{2} \cdot 6, \frac{1}{2} \cdot 6 \right) = (3, 3)
\]

### Length of Line Segment \( AB \):

Now let's calculate the length of the original line segment \( AB \) using the distance formula:

\[
\text{Length of } AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the coordinates of \( A(2, 4) \) and \( B(6, 6) \):

\[
\text{Length of } AB = \sqrt{(6 - 2)^2 + (6 - 4)^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}
\]

### Length of Line Segment \( A'B' \):

Now, we compute the length of line segment \( A'B' \):

\[
\text{Length of } A'B' = \sqrt{(3 - 1)^2 + (3 - 2)^2}
\]

Calculating the differences:

\[
\text{Length of } A'B' = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]

### Comparing the Lengths:

To prove that line segment \( A'B' \) is half the length of line segment \( AB \):

\[
\frac{1}{2} \times \text{Length of } AB = \frac{1}{2} \times 2\sqrt{5} = \sqrt{5}
\]

Since \( \text{Length of } A'B' = \sqrt{5} \), we conclude:

\[
\text{Length of } A'B' = \frac{1}{2} \text{Length of } AB
\]

Thus, line segment \( A'B' \) is indeed half the length of line segment \( AB \).