Question

arc 2 and arc 3 have the same radius.
all of the following statements must be true except for which one?
a rq = sq
b pq bisects ∠rps
c pr = pq
d pq ⊥ rs

Explanation:

Step1: Analyze each option

• Option A: Since arc 2 and arc 3 have the same radius, and \(PQ\) is constructed such that \(RQ = SQ\) (by the property of equal - radius arcs in the construction of an angle bisector).
• Option B: \(\overrightarrow{PQ}\) bisects \(\angle RPS\). In the construction of an angle bisector (using arcs of equal radius), the line from the vertex (\(P\)) to the intersection point (\(Q\)) of the arcs bisects the angle.
• Option C: \(PR = PQ\) is not necessarily True. \(PR\) is a segment from \(P\) to a point on arc 1, and \(PQ\) is a segment from \(P\) to the intersection of arc 2 and arc 3. There is no geometric principle that forces \(PR = PQ\) in this construction.
• Option D: Since \(\overrightarrow{PQ}\) is the angle bisector of \(\angle RPS\), and the construction is based on congruent triangles (by SSS - Side - Side - Side, because the radii of arc 2 and arc 3 are equal and \(PR = PS\) (radii of arc 1)), \(PQ\perp RS\) when \(PQ\) is the perpendicular bisector of \(RS\) (in the case of an isosceles triangle \(\triangle RPS\) with \(PR = PS\) and \(PQ\) as the angle bisector, it is also the perpendicular bisector).

Answer:

C. \(PR = PQ\)