A line segment has endpoints at #(5 ,8 )# and #(7 ,4)#. If the line segment is rotated about the origin by #pi #, translated horizontally by #-1 #, and reflected about the y-axis, what will the line segment's new endpoints be?

Answer 1

#(6,-8),(8,-4)#

We can create a rule for this transformation.

Assume the original endpoints of the segment can be described as #color(red)((x,y)#.
The first way in which the segment is manipulated is with a rotation of #pi#, or #180^@#. This translates by taking the opposite of the #x# and #y# values of the point, or our new "rule" for this point in the transformation: #color(red)((-x,-y)#
The endpoints are then translated horizontally by #-1#. Horizontal movement corresponds to the #x# coordinate, whereas vertical corresponds to the #y# coordinate. Since this has been horizontally shifted, the new rule, working off the most previous rule, is: #color(red)((-x-1,-y)#
The final transformation is a reflection over the #y# axis, which means that the #x# coordinate's sign is flipped: #color(red)((x+1,-y)#
This is the rule for the entire transformation. Apply it to the points #(5,8)# and #(7,4)#.
#(5,8)rarr(6,-8)#
#(7,4)rarr(8,-4)#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

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

  1. Rotation about the origin by (\pi): Rotating a point ((x, y)) around the origin by (\pi) radians (180 degrees) gives the new points ((-x, -y)). Applying this to the given endpoints:

    • (A(5, 8)) becomes (A'(-5, -8))
    • (B(7, 4)) becomes (B'(-7, -4))
  2. Translation horizontally by -1: Shifting a point ((x, y)) left by 1 unit results in the new points ((x-1, y)). Applying this to the points after rotation:

    • (A'(-5, -8)) becomes (A''(-6, -8))
    • (B'(-7, -4)) becomes (B''(-8, -4))
  3. Reflection about the y-axis: Reflecting a point ((x, y)) over the y-axis gives new points ((-x, y)). Applying this to the points after translation:

    • (A''(-6, -8)) becomes (A'''(6, -8))
    • (B''(-8, -4)) becomes (B'''(8, -4))

Thus, the new endpoints of the line segment after all the transformations are (A'''(6, -8)) and (B'''(8, -4)).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7