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If a chord \[AB\] subtends an angle \[{60^ \circ }\]at the centre of a circle, then then angle between the tangents at \[A\] and \[B\] is also \[{60^ \circ }\].

A) True

B) False

Answer
Verified

It is given that: It is given that, \[PA\] and \[PB\] are tangents to the circle with centre \[O\] at \[A\,\&\, B,\]respectively.

The chord \[AB\] subtends an angle \[{60^ \circ }\] at \[O\].

We have to find the value of the angle \[\angle APB = {60^ \circ }\] is true or false.

Now, \[PA\] and \[PB\] are tangents to the circle with centre \[O\] at \[A\,\&\, B,\]respectively. We know that the radius of a circle is perpendicular to the tangent at the tangent point.

Then, \[\angle PAO = \angle PBO = {90^ \circ }\].

By angle sum property we know that, the sum of all the angles of any quadrilaterals is \[{360^ \circ }.\]

So, from the quadrilateral \[APBO\] we get,

\[\angle PAO + \angle PBO + \angle AOB + \angle APB = {360^ \circ }\]

Substitute the values of the angles we get,

\[{90^ \circ } + {90^ \circ } + {60^ \circ } + \angle APB = {360^ \circ }\]

Simplifying we get,

\[\angle APB = {360^ \circ } - {240^ \circ }\]

We get, \[\angle APB = {120^ \circ }\]

So, the given statement is wrong.