Question

Prove that angle in a semi circle is a right angle.

Answer 1

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 Alpha and 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°

Answer 2

AnswEr:

Theorem : The angle in a semi-circle is a right angle.

$\underline\mathfrak{Given:-}$

PQ is a diameter of a circle C(O,r) and $\angle$PRQ is an angle in semi-circle.

$\underline\mathfrak{To\:Prove:-}$

$\angle$ PRQ = 90°.

___________________________

$\underline\mathfrak{Proof:-}$

We know that 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. So, we have

$\qquad\sf{\angle\:POQ=2\angle\:PRQ}$

$\Rightarrow$$\sf{180\degree=2\angle\:PRQ}$$\qquad\tt{\therefore\:[POQ\:is\:a\:straight\:line]}$

$\Rightarrow$$\sf{\angle\:PRQ=90°}$

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