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A, B and C are three points on the circle with centre O. The tangent at C meets BA produced at T. Given that, \[\angle ATC = {36^ \circ }\] and \[\angle ACT = {48^ \circ }\]. Calculate the angle subtended by AB at the centre of the circle?

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Last updated date: 13th Jun 2024
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Answer
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Hint:Here we have to find the angle subtended by AB at the center of the circle, for finding the angle we should form a triangle ACT. With help of the triangle we will find the angle in the straight line BT. With the help of the angle found on the circumference of the circle we will find the required angle.

Complete step-by-step answer:

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It is given that; A, B and C are three points on the circle with center O. The tangent at C meet BA produced at T. Given that, \[\angle ATC = {36^ \circ }\] and \[\angle ACT = {48^ \circ }\]
We have to find the value of the angle \[BOA.\]
We know that the sum of all the angles of a triangle is \[{180^ \circ }\].
From, the \[\Delta ACT\], we get,
\[\angle ACT + \angle CTA + \angle CAT = {180^ \circ }\]
Substitute the values of the given angles we get,
\[{36^ \circ } + {48^ \circ } + \angle CAT = {180^ \circ }\]
Simplifying we get,
\[\angle CAT = {96^ \circ }\]
Here, \[\angle CAB\,\&\, \angle CAT\] lies on the same line \[BT\]. So, we have,
\[\angle CAB + \angle CAT = {180^ \circ }\]
Substitute the value we get,
\[\angle CAB = {180^ \circ } - {96^ \circ } = {84^ \circ }\]
The alternate segment theorem (also known as the tangent-chord theorem) states that in any circle, the angle between a chord and a tangent through one of the end points of the chord is equal to the angle in the alternate segment.
By the alternate segment theorem, we get, \[\angle ABC = \angle ACT = {48^ \circ }\]
Again, from, the \[\Delta ABC\] we get,
\[\angle ABC + \angle BCA + \angle CAB = {180^ \circ }\]
Substitute the values of the given angles we get,
\[{84^ \circ } + {48^ \circ } + \angle BCA = {180^ \circ }\]
Simplifying we get,
\[\angle BCA = {48^ \circ }\]
We know that the angle at the centre is twice the angle at the circumference of a circle.
Here we have \[\angle BOA\] as the angle in the centre of circle and \[\angle BCA\] is the angle at circumference of the circle, hence by the above statement we get,
\[\angle BOA = 2\angle BCA\]
Substitute the value \[\angle BCA\] we get,
\[\angle BOA = 2 \times {48^ \circ } = {96^ \circ }\]
Hence, the value of the angle \[\angle BOA = {96^ \circ }\].

Note:The angle sum property states that the sum of all the angles of a triangle is \[{180^ \circ }\].The alternate segment theorem (also known as the tangent-chord theorem) states that in any circle, the angle between a chord and a tangent through one of the end points of the chord is equal to the angle in the alternate segment.We know that the angle at the centre is twice the angle at the circumference of a circle.Students should remember these theorems and properties of circle for solving these types of questions.