Can you find the radius of the semicircle?

Added: 2026-05-12

[00:00:00]Can you find the radius of the semicircle?

[00:00:09]At point C, this angle is 60°.

[00:00:18]This angle is 60°.

[00:00:22]So, this angle it will be 180° 60° 60°.

[00:00:35]That will be 60°.

[00:00:40]This angle is 60°.

[00:00:45]And now, if we complete the circle, and extend BC, suppose this point is P, then we know that opposite angles are equal.

[00:01:22]So, this angle it will be also 60°.

[00:01:32]And now, PC it will be equal to AC.

[00:01:40]That will be five.

[00:01:43]By symmetry, PC is five.

[00:01:57]And if we join AP then in triangle SCP it is a C P SC is equal to PC that is five that means these two angles they will be equal suppose alpha and this angle is 60 plus 60 that will be 120° so alpha plus alpha plus 120° is 180° that means alpha will be 30° this angle is 30° and this angle is 30° and now suppose the center of this circle is O then in any circle if AB is a chord and if it is making an angle theta on the circle, suppose this point is P then at the center O it will make an angle of 2 theta.

[00:04:34]So, here if we join AO and BO then angle AOB it will be two times angle APB and it is two times angle APB is 30° that will be 60° so this angle is 60° and AO and BO is a radius suppose X AO is X BO is X and if we join AB then then in triangle AOB It is A O B This angle is 60° and AO is X BO is X That means these two angles they will be also equal and 60° So AB it will be also X We get AB is X And now in triangle ACB It is A C B AB is X AC is 5 BC is 8 and this angle is 60° So X is squared It will be 5 is squared plus 8 is squared - 2 * 5 * 8 * cos 60° And X is squared is 25 plus 64 - 2 * 5 * 8 And two will get cancelled.

[00:07:50]So, x is squared is 25 plus 64 minus 40.

[00:08:00]That will be 49.

[00:08:04]So, we get x is squared is 49.

[00:08:09]That means x is seven.

[00:08:16]And a radius is x.

[00:08:27]That is seven.

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