Question

In the given figure, two circles intersect at A and B. The centre of the smaller circle is Oand it lies on the circumference of the larger circle. If ∠APB = 70°, find ∠ACB.

Answer 1

Consider the smaller circle whose centre is given as ‘O’.

The angle subtended by an arc at the centre of the circle is double the angle subtended by the arc in the remaining part of the circle.

So, here we have

  `angle AOB = 2 angle APB `

                = 2 (70°)

   `angle AOB = 140°`

Now consider the larger circle and the points ‘A’, ‘C’, ‘B’ and ‘O’ along its circumference. ‘ACBO’ form a cyclic quadrilateral.

In a cyclic quadrilateral it is known that the opposite angles are supplementary, meaning that the opposite angles add up to 180°.

`angleAOB + angle ACB` = 180°

                 `angle ACB = 180° - angle AOB`

                               `= 180° - 140°`

                   `angle ACB = 40°`

Hence ,the measure of`  ` angle ACB `  is 40° .