Question 2 of 11intermediate🔧 ApplyMCQ1 mark

A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is:

Exercise: EXERCISE 10.1 | Q: 3 | (Page 4)
For More Understanding

Explanation

Option D is correct. According to Theorem 10.1 referenced in the context, the angle between the radius and tangent at the point of contact is a right angle. Therefore, ∠OPQ = 90°. This forms a right triangle OPQ where OP is the radius (5 cm), OQ is the hypotenuse (12 cm), and PQ is the tangent whose length we need to find. Using the Pythagoras Theorem as shown in the context's Remarks section: PQ² = OQ² - OP² = 12² - 5² = 144 - 25 = 119, which gives PQ = 119\sqrt{119} cm.

Solution Steps

  1. Step 1: Identify that PQ is a tangent at point P and OP is the radius through the point of contact.

  2. Step 2: Apply Theorem 10.1 which states that the angle between the radius and tangent at the point of contact is 90°. So, ∠OPQ = 90°.

  3. Step 3: Recognize that triangle OPQ is a right triangle with OP = 5 cm (radius), OQ = 12 cm (hypotenuse), and PQ as the unknown side.

  4. Step 4: Apply Pythagoras Theorem: OQ² = OP² + PQ²

  5. Step 5: Substitute values: 12² = 5² + PQ²

  6. Step 6: Solve: 144 = 25 + PQ², therefore PQ² = 144 - 25 = 119

  7. Step 7: Take square root: PQ = 119\sqrt{119} cm

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