Question

step 1: copy a segment and an angle.
a) which step in the construction of copying a line segment ensures that the new line segment has the same length as the original line segment?
measurement
b) explain how you could use the construction tool or a compass and straightedge to create a line segment that is twice as long as ab.

Explanation:

Step1: Answer part a)

When copying a line segment using a compass and straight - edge, the step that ensures the new line segment has the same length as the original is: Using the compass to measure the length of the original segment (by placing the compass at the two endpoints of the original segment) and then transferring that measurement (by keeping the compass width the same and marking the new segment).

Step2: Answer part b)

1. First, copy the line segment \(\overline{AB}\) using the compass - and - straight - edge construction method. Let's call the copied segment \(\overline{CD}\) where \(C\) is a new point.
2. Then, place the compass at point \(D\). Open the compass to the length of \(\overline{AB}\) (the same length used to copy \(\overline{AB}\) initially).
3. Draw an arc from point \(D\) such that it intersects the extension of the line containing \(\overline{CD}\). Let the intersection point be \(E\). The line segment \(\overline{CE}\) is twice as long as \(\overline{AB}\) since \(CE=CD + DE\) and \(CD = AB\), \(DE=AB\).

Answer:

a) Using the compass to measure the length of the original segment and then transferring that measurement.
b) Copy the segment \(\overline{AB}\) to get a segment \(\overline{CD}\). Then, using the length of \(\overline{AB}\) (with the compass), extend from the end - point \(D\) of \(\overline{CD}\) to create a new segment \(\overline{DE}\) of length \(AB\). The segment \(\overline{CE}\) (\(C\) to \(E\)) is twice as long as \(\overline{AB}\).