If
�
�
‾
=
AB
=
10
10 inches then find the distance of
�
′
�
′
‾
A
′
B
′
after a reflection over the x-axis and a rotation of
20
°
20° clockwise.
After reflecting over the x-axis, the point A' becomes -A' (with the same y-coordinate) and the point B' becomes -B' (with the same y-coordinate). So, A' = (-10, 0) and B' = (-10, 0).
After rotating 20° clockwise, we can use the rotation matrix:
[ cos(20°) -sin(20°) ]
[ sin(20°) cos(20°) ]
To find the new coordinates of A' and B', we can multiply their original coordinates by this rotation matrix.
A' = [ cos(20°) -sin(20°) ] * (-10, 0)
= [-10 cos(20°), 10 sin(20°)]
B' = [ cos(20°) -sin(20°) ] * (-10, 0)
= [-10 cos(20°), 10 sin(20°)]
So the distance between A' and B' after the reflection over the x-axis and rotation of 20° clockwise is:
|A'B'| = sqrt(((-10 cos(20°)) - (-10 cos(20°)))^2 + ((10 sin(20°)) - (10 sin(20°)))^2)
= sqrt(0^2 + 0^2)
= 0
Therefore, the distance between A' and B' is 0.