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ICSE Class 10 Mathematics Question 30 of 49

Multiple-Choice Questions — Question 30

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Question 30

In the given diagram, O is the center of the circle, and PQ is a tangent at A. If ∠ABC = 50°, then values of x, y and z respectively are :

  1. 50°, 100°, 40°

  2. 50°, 50°, 65°

  3. 40°, 80°, 50°

  4. 50°, 25°, 78°

In the given diagram, O is the center of the circle, and PQ is a tangent at A. If ∠ABC = 50°, then values of x, y and z respectively are : Maths Competency Focused Practice Questions Class 10 Solutions.
Answer

We know that,

The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

∴ ∠AOC (y) = 2∠ABC = 2 × 50° = 100°.

In △ AOC,

⇒ OA = OC (Radii of same circle)

⇒ ∠OAC = ∠OCA = z (Angle opposite to equal sides are equal)

By angle sum property of triangle,

⇒ ∠OAC + ∠OCA + ∠AOC = 180°

⇒ z + z + 100° = 180°

⇒ 2z = 180° - 100°

⇒ 2z = 80°

⇒ z = 80°2\dfrac{80°}{2} = 40°.

We know that,

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

⇒ ∠CAQ (x) = ∠OAQ - ∠OAC = 90° - 40° = 50°.

Hence, Option 1 is the correct option.