If
�
�
‾
=
AB
=
19
19 inches then find the distance of
�
′
�
′
‾
A
′
B
′
after a reflection over the x-axis and a rotation of
293
°
293° clockwise.
�
′
�
′
‾
=
A
′
B
′
=
inche
To find the distance of A' B' after the reflection over the x-axis and rotation of 293° clockwise, we need to find the image of A' B' after each transformation.
1. Reflection over the x-axis: This will flip the sign of the y-coordinate. So, A' will become A'' with coordinates (19, -19).
2. Rotation of 293° clockwise: We can find the new coordinates using the rotation matrix:
[x' y'] = [cosθ -sinθ] * [x y]
In this case, θ = 293° and the coordinates of A'' are (19, -19):
[x' y'] = [cos293° -sin293°] * [19 -19]
Using a calculator or trigonometric tables, we can find:
cos293° = -0.3420
sin293° = -0.9397
[x' y'] ≈ (-0.3420 * 19) + (-0.9397 * -19) ≈ -12.9538 + 17.8463 ≈ 4.8925
(-0.3420 * -19) + (-0.9397 * 19) ≈ 6.4983 + 17.8463 ≈ 24.3446
Therefore, A' B' becomes A'' B'' with coordinates (4.8925, 24.3446).
To find the distance between A'' and B'', we can use the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((4.8925 - 0)^2 + (24.3446 - 0)^2)
= sqrt(23.9313 + 594.0972)
≈ sqrt(618.0285)
≈ 24.85 inches
Therefore, the distance of A'' B'' is approximately 24.85 inches.