Question

prove that an angle in a semicircle is a right angle.

Answer 1

given:- PQ is a dimeter of a circle C (O,r) and angle PRQ is an angle in a semi circle.

to proof :- angle PRQ is a right angle ie; angle PRQ = 900

proof:- The angle subtended by an arc of a circle at its centre is twice the angle formed by the same arc at a point on the circle [NCERT THEOREM 10.8]

===> angle POQ = 2 angle PRQ.

===> 1800 = 2 angle PRQ [ angle POQ is 1800 bcoz POQ is a straight line]

===> angle PRQ = 1800 / 2

===> angle PRQ = 900.

HENCE PROOVED.

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Answer 2

Proof : 
Label the diameter endpoints A and B, the top point C and the middle of the circle M.
Label the acute angles at A and B Alphaand Beta.
Draw a radius 'r' from the (right) angle point C to the middle M.

Angle MAC = ACM = Alpha because the left subtriangle is iscosceles because the opposite sides AM and CM are both radii.

Angle MBC = BCM = Beta because the right subtriangle is iscosceles because the opposite sides BM and CM are both radii.

Add up the angles at A, B and C.

This gives 2 * Alpha + 2 * Beta, which sum to 180° because ABC is a triangle.

Halving this gives Alpha + Beta (= the angle ACB) = 90°

hence proved



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