Question

In the given figure, P and Q are the mid-points of arcs
AB and CA respectively. Prove that AX = AY.
​

Answer 1

Given :- In the given figure, P and Q are the mid-points of arcs AB and CA respectively. Prove that AX = AY. ?

Solution :-

Construction :-

• Join AP and AQ .

given that, P is the Mid - Point of arc AB .

so,

→ arc AP = arc PB

then,

→ ∠AQP = ∠PAB = ɑ (Let) { Angles made in the same segment by same arc is equal. }

similarly,

given that, Q is the Mid - Point of arc CA .

so,

→ arc AQ = arc QC

then,

→ ∠QPA = ∠QAC = θ (Let) { Angles made in the same segment by same arc is equal. }

then,

• In ∆AXY => ∠AXY = ɑ + θ { Exterior angle for ∆APX is equal to sum of opposite interior angles. }
• In ∆AYX => ∠AYX = ɑ + θ { Exterior angle for ∆AQY is equal to sum of opposite interior angles. }

therefore,

→ ∠AXY = ɑ + θ = ∠AYX

hence,

→ AX = AY . { Sides Opposite to equal angles are equal in Length. } [ Hence, Proved. ]

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