A line segment has endpoints at #(1 ,2 )# and #(5 , 8 )#. If the line segment is rotated about the origin by #(3 pi ) /2 #, translated vertically by # -3 #, and reflected about the y-axis, what will the line segment's new endpoints be?

Question

A line segment has endpoints at #(1  ,2   )# and #(5   , 8   )#.  If the line segment is rotated about the origin by #(3  pi ) /2 #, translated vertically by # -3  #, and reflected about the y-axis, what will the line segment's new endpoints be?

Theodore Abad · Accepted Answer

#(-2,-4)" and " (-8,-8)# #"Since there are 3 transformations to be performed name the "# #"endpoints A(1,2) and B(5,8)"# #color(blue)"First transformation"# #"under a rotation about the origin of "(3pi)/2# #• " a point "(x,y)to(y,-x)# #rArrA(1,2)toA'(2,-1)# #rArrB(5,8)toB'(8,-5)# #color(blue)"Second transformation"# #"under a dilatation of "((0),(-3))# #• " a point "(x,y)to(x,y-3)# #rArrA'(2,-1)toA''(2,-4)# #rArrB'(8,-5)toB''(8,-8)# #color(blue)"Third transformation"# #"under a reflection in the y-axis"# #• " a point "(x,y)to(-x,y)# #rArrA''(2,-4)toA'''(-2,-4)# #rArrB''(8,-8)toB'''(-8,-8)# #"after all 3 transformations"# #(1,2)to(-2,-4)" and "(5,8)to(-8,-8)#

Carson Arnold · Answer

undefined To find the new endpoints of the line segment after the specified transformations, follow these steps:

1. Rotate the original endpoints by $\frac{3\pi}{2}$ radians about the origin.
2. Translate the rotated endpoints vertically by -3 units.
3. Reflect the translated endpoints about the y-axis.

Let's start with the original endpoints:

Endpoint 1: (1, 2)
Endpoint 2: (5, 8)

1. Rotate the original endpoints by $\frac{3\pi}{2}$ radians about the origin:
Endpoint 1 after rotation: $(2, -1)$
Endpoint 2 after rotation: $(8, -5)$

2. Translate the rotated endpoints vertically by -3 units:
Endpoint 1 after translation: $(2, -4)$
Endpoint 2 after translation: $(8, -8)$

3. Reflect the translated endpoints about the y-axis:
Endpoint 1 after reflection: $(-2, -4)$
Endpoint 2 after reflection: $(-8, -8)$

So, the new endpoints of the line segment after the specified transformations are:
Endpoint 1: $(-2, -4)$
Endpoint 2: $(-8, -8)$