Home > Geometry > Rigid Transformations

A line segment has endpoints at #(2 ,3 )# and #(3 ,9 )#. If the line segment is rotated about the origin by #( pi)/2 #, translated vertically by #-8 #, and reflected about the x-axis, what will the line segment's new endpoints be?

Answer 1

Allison Allen

#(2,3)to(3,6)" and " (3,9)to(-9,5)#

Since there are 3 transformations to be performed here, label the endpoints A (2 ,3) and B (3 ,9)

First Transformation Under a rotation about the origin of #pi/2#

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

Hence A(2 ,3) → A'(-3 ,2) and B(3 ,9) → B' (-9 ,3)

Second Transformation Under a translation #((0),(-8))#

#" a point " (x,y)to(x,y-8)#

Hence A'(-3 ,2) → A''(3 ,-6) and B'(-9 ,3) → B''(-9 ,-5)

Third transformation Under a reflection in the x-axis

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

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

Thus after all 3 transformations.

#(2,3)to(3,6)" and " (3,9)to(-9,5)#

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

Answer 2

Claire Akin

To find the new endpoints of the line segment after it's rotated about the origin by π/2, translated vertically by -8, and reflected about the x-axis, follow these steps:

Rotate about the origin by π/2: To rotate a point (x, y) by angle θ counterclockwise about the origin, the new coordinates (x', y') can be found using the formulas: x' = x * cos(θ) - y * sin(θ) y' = x * sin(θ) + y * cos(θ)

Applying these formulas to the endpoints (2, 3) and (3, 9) with θ = π/2: For (2, 3): x' = 2 * cos(π/2) - 3 * sin(π/2) = -3 y' = 2 * sin(π/2) + 3 * cos(π/2) = 2 For (3, 9): x' = 3 * cos(π/2) - 9 * sin(π/2) = -9 y' = 3 * sin(π/2) + 9 * cos(π/2) = 3
Translate vertically by -8: To translate a point vertically, subtract the translation value from the y-coordinate. New y-coordinates: y' = y - 8

Applying this to the rotated points: For (-3, 2): y' = 2 - 8 = -6 For (-9, 3): y' = 3 - 8 = -5
Reflect about the x-axis: To reflect a point about the x-axis, change the sign of the y-coordinate. New y-coordinates: y' = -y

Applying this to the translated points: For (-3, -6): y' = -(-6) = 6 For (-9, -5): y' = -(-5) = 5

So, the new endpoints of the line segment after the transformations are (-3, 6) and (-9, 5).

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

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.