Problem

Use the displayed perpendicular-bisector construction to identify the midpoint of \(AB\) and justify it from the point's incidence and equal distances to \(A\) and \(B\).

Construction arcs intersect at U and V; line UV intersects segment AB at labeled point M.

Big Picture

What this problem is really about

A construction’s auxiliary points are evidence, not automatically the requested point. We’ll use the equal-radius arcs to establish an equidistant locus, identify the line they determine, and then focus on where that line crosses the original segment. Incidence on the segment plus equal endpoint distances is the test that isolates its midpoint.

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Preview only

Point M is the midpoint of AB because it is where the perpendicular bisector meets AB, so MA = MB.

Point A is the midpoint of AB because the perpendicular bisector is constructed from the endpoints of the segment.

The midpoint is the point halfway between the two arc intersections because the perpendicular bisector is centered there.

Any point on the perpendicular bisector is the midpoint of AB because points on that line are equidistant from A and B.

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Use the displayed perpendicular-bisector construction to identify the midpoint of \(AB\) and justify it from the point's incidence and equal distances to \(A\) and \(B\).

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