A line segment has endpoints at #(7 ,4 )# and #(3 ,5 )#. If the line segment is rotated about the origin by #(3 pi)/2 #, translated vertically by #-2 #, and reflected about the y-axis, what will the line segment's new endpoints be?

Question

A line segment has endpoints at #(7  ,4    )# and #(3  ,5  )#.  If the line segment is rotated about the origin by #(3   pi)/2   #, translated vertically by #-2  #, and reflected about the y-axis, what will the line segment's new endpoints be?

Aurora Ayala · Accepted Answer

#(7,4)to(-4,-9)" and " (3,5)to(-5,-5)#

Since there are 3 transformations to be performed, name the endpoints A(7 ,4) and B(3 ,5) so that we can 'track' the coordinates of the endpoints after each transformation.

First transformation Under a rotation about the origin of #(3pi)/2#

a point (x ,y) → (y ,-x)

hence A(7 ,4) → A'(4 ,-7) and B(3 ,5) → B'(5 ,-3)

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

a point (x ,y) → (x ,y-2)

hence A'(4 ,-7) → A''(4 ,-9) and B'(5 ,-3) → B''(5 ,-5)

Third transformation Under a reflection in the y-axis

a point (x ,y) → (-x ,y)

hence A''(4 ,-9) → A'''(-4 ,-9) and B''(5 ,-5) → B'''(-5 ,-5)

Thus after all 3 transformations the endpoints are.

#(7,4)to(-4,-9)" and " (3,5)to(-5,-5)#

Jayden Jackson · Answer

undefined To find the new endpoints of the line segment after the described transformations:

1. **Rotation about the origin by $ \frac{3\pi}{2} $**:
   Applying the rotation matrix:
   $$
   \begin{bmatrix}
   \cos\left(\frac{3\pi}{2}\right) & -\sin\left(\frac{3\pi}{2}\right) \
   \sin\left(\frac{3\pi}{2}\right) & \cos\left(\frac{3\pi}{2}\right)
   \end{bmatrix}
   \begin{bmatrix}
   x \
   y
   \end{bmatrix}
   $$
   This simplifies to:
   $$
   \begin{bmatrix}
   0 & -1 \
   1 & 0
   \end{bmatrix}
   \begin{bmatrix}
   x \
   y
   \end{bmatrix}
   $$
   For each endpoint, apply this transformation.

2. **Translation vertically by -2**:
   Subtract 2 from the y-coordinate of each endpoint.

3. **Reflection about the y-axis**:
   Negate the x-coordinate of each endpoint.

Now, apply these transformations to the original endpoints:

1. For the endpoint (7, 4):
   - Rotation: $\begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} 7 \ 4 \end{bmatrix} = \begin{bmatrix} -4 \ 7 \end{bmatrix}$
   - Translation: $( -4, 7 - 2) = (-4, 5)$
   - Reflection: $(-(-4), 5) = (4, 5)$

2. For the endpoint (3, 5):
   - Rotation: $\begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} 3 \ 5 \end{bmatrix} = \begin{bmatrix} -5 \ 3 \end{bmatrix}$
   - Translation: $(-5, 3 - 2) = (-5, 1)$
   - Reflection: $(-(-5), 1) = (5, 1)$

Therefore, the new endpoints of the line segment after the described transformations are $(4, 5)$ and $(5, 1)$.