Question) In the given figure, P is the mid-point of arc APB and M is the mid-point of chord AB - Maths - Circles

Question

Question) In the given figure, P is the mid-point of arc APB and M
is the mid-point of chord AB of a circle with centre O Prove that
:

(i) P M ⊥ A B
(ii) PM produced will pass through the centre O;

(iii) PM produced will bisect the major arc AB

Neha Sethi · Accepted Answer

Dear student
We have,arcAP=arc PB⇒AP=PB   [Chords of congruent arcs are equal]⇒OP is the bisector of ∠APB⇒∠OPA=∠OPBThus, in△AMP and △BMP, we have,PM=PM [Common]AP=PB∠OPA=∠OPBSo, by SAS congruence criterion, we have△AMP≅△BMP⇒AM=BM and ∠AMP=∠BMPBut ∠AMP+∠BMP=180°∴∠AMP=∠BMP=90°Hence PM or PO is the prependicular bisector of AB
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Question) In the given figure, P is the mid-point of arc APB and M is the mid-point of chord AB of a circle with centre O. . Prove that : (i) P M ⊥ A B (ii) PM produced will pass through the centre O; (iii) PM produced will bisect the major arc AB.

Question) In the given figure, P is the mid-point of arc APB and M is the mid-point of chord AB of a circle with centre O. . Prove that :

(i) $P M ⊥ A B$
(ii) PM produced will pass through the centre O;

(iii) PM produced will bisect the major arc AB.