A line segment has endpoints at #(2 ,7 )# and #(5 ,4 )#. The line segment is dilated by a factor of #2 # around #(4 ,3 )#. What are the new endpoints and length of the line segment?

Question

A line segment has endpoints at #(2  ,7  )# and #(5 ,4   )#.  The line segment is dilated by a factor of #2  # around #(4  ,3  )#.  What are the new endpoints and length of the line segment?

Sadie Allen · Accepted Answer

#color(violet)("New end points are " (0,110, (6,5) " and the length of the line segment is " 8.49# #A(2,7), B(5,4), " about "C(4,3) " & dilated by factor " 2# #A'((x),(y)) = 2a -c = 2 * ((2),(7)) - ((4),(3)) = ((0), (11))# #B'((x),(y)) = 2b -c = 2 * ((5),(4)) - ((4),(3)) = ((6), (5))# #"Length of the line segment after dilation " = vec(A'B')# #vec (A'B') = sqrt((0-6)^2 = (11-5)^2) = 8.49#

Wyatt Anderson · Answer

undefined To find the new endpoints of the line segment after dilation by a factor of 2 around the point (4, 3), we first find the new coordinates of each endpoint using the dilation formula:

For a point (x, y) dilated by a factor of k around the point (h, v), the new coordinates (x', y') are given by:

$$ x' = h + k(x - h) $$
$$ y' = v + k(y - v) $$

Using these formulas, we can find the new coordinates of both endpoints.

For the first endpoint (2, 7):

$$ x' = 4 + 2(2 - 4) = 0 $$
$$ y' = 3 + 2(7 - 3) = 11 $$

So, the new coordinates of the first endpoint are (0, 11).

For the second endpoint (5, 4):

$$ x' = 4 + 2(5 - 4) = 6 $$
$$ y' = 3 + 2(4 - 3) = 5 $$

So, the new coordinates of the second endpoint are (6, 5).

Now, we can calculate the length of the new line segment using the distance formula:

$$ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

For the new endpoints (0, 11) and (6, 5):

$$ \text{Length} = \sqrt{(6 - 0)^2 + (5 - 11)^2} $$
$$ \text{Length} = \sqrt{36 + 36} $$
$$ \text{Length} = \sqrt{72} $$

So, the new endpoints of the line segment are (0, 11) and (6, 5), and the length of the new line segment is $ \sqrt{72} $.