Question

The diameters PQ and RS of the circle with centre C are perpendicular to each other at C. State, why arc PS and arc SQ are congruent. Write the other arcs which are congruent to arc PS.

Answer 1

diameter PQ ⊥ diameter RS … [Given]

∴ m∠PCS = m∠SCQ = m∠PCR = m∠RCQ = 90°

The measure of the angle subtended at the centre by an arc is the measure of the arc.

∴ m(arc PS) = m∠PCS = 90° …(i) 

m (arc SQ) = m∠SCQ = 90° …(ii)

∴ m(arc PS) = m(arc SQ) … [From (i) and (ii)]

∴ arc PS ≅ arc SQ … [If the measures of two arcs of a circle are same, then the two arcs are congruent]

m(arc PR) = m∠PCR = 90° ...(iii) 

m (arc RQ) = m∠RCQ = 90° … (iv)

∴ m(arc PS) = m(arc PR) = m(arc RQ) … [From (i), (iii) and (iv)]

∴ arc PS ≅ arc PR ≅ arc RQ

… [If the measures of two arcs of a circle are same, then the two arcs are congruent]

∴ arc PR and arc RQ are congruent to arc PS.