Question

A line segment has endpoints at `(1 ,4 )` and `(7 ,5 )`. If the line segment is rotated about the origin by `pi`, translated horizontally by `- 2`, and reflected about the x-axis, what will the line segment's new endpoints be?

Answer 1

(-3 ,4) and (-9 ,5)

Explanation:

Let's begin by naming the endpoints A(1 ,4) and B(7 ,5) so we can follow what happens to them.

Step 1

Under a rotation of `pi" about the origin "`

A point (x ,y) → (-x ,-y)

hence A(1 ,4) → A'(-1 ,-4) and B(7 ,5) → B'(-7 ,-5)

Step 2

Under a translation of `((-2),(0))`

A point (x ,y) → (x-2 ,y)

hence A'(-1 ,-4) → A''(-3 ,-4) and B'(-7 ,-5) → B''(-9 ,-5)

Step 3

Under a reflection in the x-axis

A point (x ,y) → (x ,-y)

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

After all 3 transformations:

(1 ,4) → (-3 ,4) and (7 ,5) → (-9 ,5)