Question

A segment with endpoints (5,8) and (-6,8) is rotated around the origin. How long will the new segment be?

Answer 1

`color(blue)("The same length")`

Explanation:

The simple answer to this is, it will be the same length. Rotating an image does not change its dimensions, only its position within a coordinate system.

We can show this by rotating the line segment counter-clockwise about the origin through an angle `pi/2`.

A rotation of `pi/2` counter-clockwise maps:

`(x,y)->(y,-x)`

Using given points:

`(5,8)->(8,-5)`

`(-6,8)->(8,6)`

Using the distance formula, with coordinates `(5,8) , (-6,8)`

`d=sqrt((5-(-6))^2+(8-8)^2)=11`

Using the distance formula, with coordinates `(8,-5) , (8,6)`

`d=sqrt((8-8)^2+(-5-6)^2)=11`

So you can see that the relative distance remain unchanged.

Answer 2

The segment is clearly eleven units long before rotation, and rotation won't change that, so `11`.