A line segment has endpoints at #(5 ,5 )# and #(3 ,5 )#. The line segment is dilated by a factor of #1/2 # around #(3 , 2)#. What are the new endpoints and length of the line segment?

Question

A line segment has endpoints at #(5  ,5  )# and #(3  ,5   )#.  The line segment is dilated by a factor of #1/2  # around #(3 , 2)#.  What are the new endpoints and length of the line segment?

James Atkins · Accepted Answer

The new endpoints are #(4,7/2)# and #(3,7/2)# and the length of the line segment is #=1#

Let the line segment be #AB#

#A=(5,5)# and

#B=(3,5)#

#C=(3,2)#

The length of the segment #AB=2#

Let point #A'=(x,y)# be the point after dilatation, then

#vec(CA')=1/2 vec(CA)#

#((x-3),(y-2))=1/2((5-3),(5-2))=1/2((2),(3))#

Therefore,

#x-3=1#, #=>#, #x=4#

#y-2=3/2#, #=>#, #y=7/2#

The point #A'=(4,7/2)#

Let point #B'=(p,q)# be the point after dilatation, then

#vec(CB')=1/2 vec(CB)#

#((p-3),(q-2))=1/2((3-3),(5-2))=1/2((0),(3))#

Therefore,

#p-3=0#, #=>#, #p=3#

#q-2=3/2#, #=>#, #q=7/2#

The point #B'=(3,7/2)#

The length of the segment #A'B'=1#

Sadie Adams · Answer

undefined To find the new endpoints and length of the line segment after dilation by a factor of $ \frac{1}{2} $ around the point $ (3, 2) $, follow these steps:

1. Find the coordinates of the midpoint of the original line segment.
2. Apply the dilation transformation to each endpoint of the line segment using the given factor and midpoint as the center.
3. Calculate the distance between the new endpoints to find the length of the dilated line segment.

Let's proceed with the calculations:

1. Midpoint of the original line segment:
$$ M_x = \frac{5 + 3}{2} = 4 $$
$$ M_y = \frac{5 + 5}{2} = 5 $$
So, the midpoint is $ (4, 5) $.

2. Dilation of the endpoints:
$$ \text{New endpoint 1:} $$
$$ x_1' = 3 + \frac{1}{2}(5 - 3) = 3 + 1 = 4 $$
$$ y_1' = 2 + \frac{1}{2}(5 - 2) = 2 + \frac{3}{2} = \frac{7}{2} $$
So, the new endpoint 1 is $ (4, \frac{7}{2}) $.

$$ \text{New endpoint 2:} $$
$$ x_2' = 3 + \frac{1}{2}(3 - 3) = 3 $$
$$ y_2' = 2 + \frac{1}{2}(5 - 2) = 2 + \frac{3}{2} = \frac{7}{2} $$
So, the new endpoint 2 is $ (3, \frac{7}{2}) $.

3. Length of the dilated line segment:
$$ \text{Length} = \sqrt{(x_2' - x_1')^2 + (y_2' - y_1')^2} $$
$$ = \sqrt{(3 - 4)^2 + \left(\frac{7}{2} - \frac{7}{2}\right)^2} $$
$$ = \sqrt{(-1)^2 + 0^2} = \sqrt{1} = 1 $$

Therefore, the new endpoints of the line segment after dilation are $ (4, \frac{7}{2}) $ and $ (3, \frac{7}{2}) $, and the length of the dilated line segment is $ 1 $.