Question 7 of 11beginner🔧 ApplyShort Answer2 marks

The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

Correct Answer

Let O be the centre of the circle and let the tangent from point A touch the circle at point B. Since the tangent is perpendicular to the radius at the point of contact (Theorem 10.1), triangle OAB is a right-angled triangle with the right angle at B.

Given: OA = 5 cm (distance from centre), AB = 4 cm (tangent length), OB = r (radius). Using Pythagoras Theorem: OA² = OB² + AB², so 5² = r² + 4², giving 25 = r² + 16, therefore r² = 9.

Radius of the circle = 3 cm.

Exercise: EXERCISE 10.2 | Q: 6 | (Page 9)
For More Understanding

Explanation

This question tests the application of Theorem 10.1 from the textbook, which states that the tangent at any point of a circle is perpendicular to the radius through the point of contact. When a tangent is drawn from an external point to a circle, it forms a right-angled triangle with the radius and the line joining the external point to the centre. The Pythagoras theorem can then be applied to find the unknown dimension.

Solution Steps

  1. Step 1: Identify that the tangent from A touches the circle at B, and by Theorem 10.1, OB is perpendicular to AB.

  2. Step 2: Recognize that triangle OAB is right-angled at B, with OA as hypotenuse.

  3. Step 3: Apply Pythagoras Theorem: OA² = OB² + AB²

  4. Step 4: Substitute: 5² = r² + 4², giving 25 = r² + 16

  5. Step 5: Solve: r² = 9, therefore r = 3 cm

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