A tangent line PA is drawn from an external point P to a circle. If the radius is 8 cm and the distance from P to the center is 17 cm, what is the length of the tangent PA?

Correct — B. A radius drawn to the point of contact is perpendicular to the tangent, so △OAP has its right angle at A and OP as hypotenuse.

Pythagoras in △OAP: OP² = OA² + PA²
Rearrange: PA² = 17² − 8²
Square and subtract: 289 − 64 = 225
Take the root: PA = √225 = 15 cm → option (b)

Why the others are wrong

• (a)
  9 cm — 9 cm is 17 − 8, subtracting the lengths. Pythagoras subtracts the squares: 289 − 64 = 225, whose root is 15.
• (c)
  25 cm — 25 cm is 17 + 8. The tangent is a leg of right △OAP, so it must be shorter than the hypotenuse OP = 17 cm.
• (d)
  28 cm — 28 cm is longer than OP = 17 cm. In right △OAP the tangent PA is a leg, and a leg is always shorter than the hypotenuse.

The tangent at any point of a circle is perpendicular to the radius through that point. So the centre O, the point of contact A and the external point P form a right triangle with OP as hypotenuse.

That gives tangent length = √(OP² − r²). Here 8, 15, 17 is a Pythagorean triple, which is why the root comes out whole.

Key facts

• Radius ⟂ tangent at the point of contact.
• Tangent length from an external point P = √(OP² − r²).
• 8, 15, 17 is a Pythagorean triple: 64 + 225 = 289.

Common traps

• Subtracting the lengths (17 − 8 = 9) instead of their squares.
• Treating the tangent as the hypotenuse and adding squares: √(289 + 64) ≈ 18.8 cm.

No directly related past PYQ was found.