Question
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Write true or false and give reasons for your answers.
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 Verified
Hint:Using the given information we will try to find the value of the angle between the tangents at \[A\] and \[B\].We use the property that radius of a circle is perpendicular to the tangent at the tangent point and sum of angles of quadrilaterals is \[{360^ \circ }\].Using these properties ,we find the value of angle between tangents. If the value of the angle is \[{60^ \circ }\], the statement will be true. Otherwise, it will be false.

Complete step-by-step answer:

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.

So, the correct answer is “Option B”.

Note:The tangent is a straight line which touches any circle at a single point. The radius of a circle is perpendicular to the tangent at the tangent point.By angle sum property we know that, the sum of all the angles of any quadrilateral is \[{360^ \circ }\].