Question

A line segment has endpoints at `(5 ,9 )` and `(6 ,7 )`. The line segment is dilated by a factor of `4` around `(5 ,1 )`. What are the new endpoints and length of the line segment?

Answer 1

`(5,33)` & `(9,24)`

Explanation:

Start by drawing the line in a rectangular coordinate plane and locating the point of dilation `(5,1)`.

Draw lines from `(5,1)` through each endpoint, extending above.

Since the point `(5,1)` and `(5,9)` have a `y`-difference of `8`. Multiply this by `4` to get `32`. Add `1` to get to the point location from `(5,1)`.

The slope of the original line = `-2/1`. From the point `(5,33)`, apply this slope `4` times to find the second endpoint. It needs to lie on the line from `(5,1)` through `(6,7)`.

You can see the process below:

The dilated line segment should be `4` times bigger.

Original line segment length `= sqrt(1^2 + 2^2) = sqrt(5) ~~ 2.236`

Dilated line segment length = `sqrt(8^2 + 4^2) = sqrt(80) = sqrt(16*5) = 4sqrt(5) ~~ 8.9443`

So endpoints of the dilated line are: `(5,33)` & `(9,24)`