Question

use the compass and straightedge to construct a segment congruent to \overline{ab} with endpoint p.

Explanation:

Step1: Measure the length of segment \(AB\)

Use the compass to measure the distance between points \(A\) and \(B\). Place the compass's needle at point \(A\) and adjust the compass so that the pencil end touches point \(B\).

Step2: Draw an arc with center \(P\)

Without changing the compass width (which is set to the length of \(AB\)), place the compass's needle at point \(P\). Then draw an arc.

Step3: Mark the new point

Use the straightedge to draw a line segment from \(P\) to a point (let's call it \(Q\)) on the arc. The segment \(PQ\) is congruent to \(AB\).

Answer:

Constructed segment \(PQ\) (where \(Q\) is the appropriate point on the arc) is congruent to \(\overline{AB}\) with endpoint \(P\).