Question

If a segment has endpoints (0, 4) and (3, 4), how long will the segment be after it is rotated 270° clockwise?

Answer 1

To determine the length of the segment after it is rotated 270° clockwise, we first need to find the length of the segment itself and then analyze how rotation affects it.

1. **Find the length of the segment with endpoints (0, 4) and (3, 4)**:
The segment lies horizontally at \(y = 4\), spanning from \(x = 0\) to \(x = 3\). The length of the segment can be calculated as:
\[
Length = |x_2 - x_1| = |3 - 0| = 3
\]

2. **Rotation of the segment 270° clockwise**:
Rotating a point (x, y) 270° clockwise around the origin follows this transformation:
\[
(x', y') = (y, -x)
\]

- For the first endpoint (0, 4):
\[
(0', 4') = (4, -0) = (4, 0)
\]

- For the second endpoint (3, 4):
\[
(3', 4') = (4, -3)
\]

After the rotation, the new endpoints are (4, 0) and (4, -3).

3. **Calculating the length of the new segment**:
The new segment has endpoints (4, 0) and (4, -3). This segment is vertical at \(x = 4\) and spans from \(y = 0\) to \(y = -3\). The length can be calculated as:
\[
Length' = |y_2 - y_1| = |-3 - 0| = |-3| = 3
\]

4. **Conclusion**:
The length of the segment after rotating it 270° clockwise remains the same. Therefore, the length of the segment after the rotation is:
\[
\boxed{3}
\]