A line segment has endpoints at #(9 ,6 )# and #(5 ,3)#. If the line segment is rotated about the origin by #pi /2 #, translated vertically by #2 #, and reflected about the x-axis, what will the line segment's new endpoints be?

Question

A line segment has endpoints at #(9 ,6    )# and #(5  ,3)#.  If the line segment is rotated about the origin by #pi  /2   #, translated vertically by #2  #, and reflected about the x-axis, what will the line segment's new endpoints be?

Evelyn Arboleda · Accepted Answer

#(9,6)to(6,7) , (5,3)to(3,3)#

Since there are 3 transformations to be performed here, label the points A(9 ,6) and B(5 ,3) so that the change to each point after each transformation can be noted.

First transformation Under a rotation about the origin if #pi/2#

#" a point" (x,y)to(y,-x)#

hence A(9 ,6) →A'(6 ,-9) and B(5 ,3) → B'(3 ,-5)

Second transformation Under a translation #((0),(2))#

#" a point" (x,y)to(x+0,y+2)#

hence A'(6 ,-9) → A''(6 ,-7) and B'(3 ,-5) → B''(3 ,-3)

Third transformation Under a reflection in the x-axis

#" a point" (x,y)to(x,-y)#

hence A''(6 ,-7) → A'''(6 ,7) and B''(3 ,-3) → B'''(3 ,3)

After all 3 transformations.

#(9,6)to(6,7)" and " (5,3)to(3,3)#

Carson Arnold · Answer

undefined To find the new endpoints of the line segment after the described transformations, we need to perform each transformation step by step:

1. **Rotation by π/2 about the origin**: To rotate a point (x, y) by an angle θ about the origin, the new coordinates (x', y') can be found using the formulas:
$$x' = x \cdot \cos(\theta) - y \cdot \sin(\theta)$$
$$y' = x \cdot \sin(\theta) + y \cdot \cos(\theta)$$
Applying these formulas with θ = π/2, the original endpoints (9, 6) and (5, 3) become (-6, 9) and (-3, 5), respectively.

2. **Translation vertically by 2**: To translate a point (x, y) vertically by a distance d, the new coordinates become (x, y + d). Applying this translation, the new endpoints become (-6, 11) and (-3, 7), respectively.

3. **Reflection about the x-axis**: To reflect a point (x, y) about the x-axis, the y-coordinate becomes its negative, i.e., (x, -y). After reflection, the new endpoints become (-6, -11) and (-3, -7), respectively.

So, after the described transformations, the line segment's new endpoints are (-6, -11) and (-3, -7).