If AB = 15 inches then find the distance of A'B' after a reflection over the x-axis and a rotation of 274 clockwise

In a reflection over the x-axis, the y-coordinate of a point is negated. So for point A(0, 15), the reflected point A' will be (0, -15).

Then, in a rotation of 274 degrees clockwise, the point will rotate around the origin to a new position. Since 274 degrees is equivalent to 6 full rotations (360 degrees each) plus an additional 254 degrees, we can reduce this rotation to just 254 degrees clockwise.

To find the coordinates of A' after a rotation of 254 degrees clockwise, we can use the following formulas:

x' = x * cos(theta) - y * sin(theta)
y' = x * sin(theta) + y * cos(theta)

Applying these formulas for A' and a rotation of 254 degrees:

x' = 0 * cos(254) - (-15) * sin(254) ≈ -6.362
y' = 0 * sin(254) + (-15) * cos(254) ≈ -14.325

Therefore, the coordinates of A' after a reflection over the x-axis and a rotation of 274 degrees clockwise are approximately (-6.362, -14.325). To find the distance of A'B', we use the distance formula:

A'B' = √((-6.362)^2 + (-14.325)^2) ≈ 15.47 inches

So, the distance of A'B' after a reflection over the x-axis and a rotation of 274 degrees clockwise is approximately 15.47 inches.