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copying a segment — Definition, Formula & Examples

Copying a segment is the compass-and-straightedge procedure for creating a new line segment that has exactly the same length as a given segment, without measuring with a ruler.

Given a segment AB‾\overline{AB} and a point CC on a line ℓ\ell, to copy AB‾\overline{AB} is to construct a point DD on ℓ\ell such that CD=ABCD = AB, using only an unmarked straightedge and a compass.

How It Works

Place the compass point on one endpoint of the original segment and open it to the other endpoint, capturing the length. Without changing the compass width, move the compass point to the starting point of the new segment and draw an arc that crosses the line. The intersection of the arc and the line marks the second endpoint, producing a congruent copy of the original segment.

Worked Example

Problem: Given segment AB‾\overline{AB} with length 5 cm and a ray starting at point PP, copy AB‾\overline{AB} onto the ray.
Step 1: Place the compass point on AA and open the compass until the pencil reaches BB. The compass now spans the length of AB‾\overline{AB}.
Step 2: Without adjusting the compass width, place the compass point on PP and draw an arc that intersects the ray. Label the intersection point QQ.
Step 3: Draw PQ‾\overline{PQ}. Because the compass width was unchanged, PQ=ABPQ = AB.
PQ=AB=5 cmPQ = AB = 5\text{ cm}
Answer: PQ‾\overline{PQ} is a copy of AB‾\overline{AB}, with PQ=5PQ = 5 cm.

Why It Matters

Copying a segment is the simplest compass-and-straightedge construction and appears as a sub-step in nearly every other construction, including copying an angle, constructing an equilateral triangle, and transferring measurements in geometric proofs. Mastering it first makes every later construction faster and more accurate.

Common Mistakes

Mistake: Accidentally changing the compass width while moving it from the original segment to the new location.
Correction: Lock the compass hinge firmly before lifting it. If your compass slips easily, tighten the adjustment screw or hold the top knob steady as you reposition.

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