Question

The length of tangent from a point 17 cm away from the centre of a circle of radius 8 cm is _____ cm.

• A. 15
• B. 20
• C. 25
• D. 30

Hint:

Remember common Pythagorean triplets: \[ (3,4,5),\ (5,12,13),\ (8,15,17) \] Recognizing them helps solve geometry problems much faster.

Answer 1

The Correct Option is A

Concept: A tangent drawn from an external point to a circle is perpendicular to the radius at the point of contact. Therefore, the radius, tangent, and line joining the center to the external point form a right-angled triangle. Hence we use the Pythagoras Theorem: \[ (\text{Hypotenuse})^2 = (\text{Perpendicular})^2 + (\text{Base})^2 \]

Step 1: Identify the given quantities. Distance from center to external point: \[ OP = 17 \text{ cm} \] Radius of the circle: \[ OT = 8 \text{ cm} \] Length of tangent: \[ PT = x \]

Step 2: Understand the right triangle formed. Since radius is perpendicular to tangent: \[ OT \perp PT \] Therefore: \[ \triangle OPT \] is a right-angled triangle. Here:
• Hypotenuse \(= OP = 17\)
• One side \(= OT = 8\)
• Other side \(= PT = x\)

Step 3: Apply Pythagoras Theorem. \[ OP^2 = OT^2 + PT^2 \] Substitute values: \[ 17^2 = 8^2 + x^2 \]

Step 4: Calculate the squares. \[ 289 = 64 + x^2 \]

Step 5: Isolate \(x^2\). \[ x^2 = 289 - 64 \] \[ x^2 = 225 \]

Step 6: Take square root on both sides. \[ x = \sqrt{225} \] \[ x = 15 \] Since length cannot be negative: \[ x = 15 \text{ cm} \]

Step 7: Verification. The values: \[ 8,\ 15,\ 17 \] form a well-known Pythagorean triplet: \[ 8^2 + 15^2 = 17^2 \] Hence answer is correct. Final Answer: \[ \boxed{15 \text{ cm}} \]