Question 3 of 11intermediate🔧 ApplyMCQ1 mark

From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is

Exercise: EXERCISE 10.2 | Q: 1 | (Page 8)
For More Understanding

Explanation

Option (A) is correct. The provided context states in the proof of Theorem 10.2 that the angle between the radius and the tangent at the point of contact is a right angle. Consequently, the line segment joining the centre to the external point, the radius, and the tangent form a right-angled triangle. We can apply the Pythagoras Theorem to find the radius, as demonstrated in the Remarks section of the text.

Solution Steps

  1. Step 1: Let the centre of the circle be O and the point of contact be P. The length of the tangent QP is 24 cm, and the distance of Q from the centre OQ is 25 cm.

  2. Step 2: Since the radius is perpendicular to the tangent at the point of contact, angle ∠OPQ\angle OPQ is 90∘90^\circ. Thus, △OPQ\triangle OPQ is a right-angled triangle.

  3. Step 3: Apply the Pythagoras Theorem in △OPQ\triangle OPQ: OQ2=OP2+QP2OQ^2 = OP^2 + QP^2.

  4. Step 4: Substitute the known values: (25)2=OP2+(24)2(25)^2 = OP^2 + (24)^2.

  5. Step 5: Solve for the radius OP: 625=OP2+576625 = OP^2 + 576, which gives OP2=625−576=49OP^2 = 625 - 576 = 49.

  6. Step 6: Taking the square root, OP=7OP = 7 cm. The radius is 7 cm.

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