Question

A point P is at a distance of 13 cm from the centre of a circle. If the radius of the circle is 5 cm, the length of the tangent from P to the circle is

• A. 12 cm
• B. 13 cm
• C. 15 cm
• D. 18 cm

Hint:

For a tangent drawn from an external point, the radius at the point of contact is always perpendicular to the tangent. So, use the Pythagoras theorem directly.

Answer 1

The Correct Option is A

Step 1: Understand the geometry of the figure.}
Let $O$ be the centre of the circle and let $PT$ be the tangent from external point $P$ touching the circle at point $T$. The radius $OT$ is perpendicular to the tangent $PT$. Therefore, triangle $OPT$ is a right-angled triangle.

Step 2: Write the given values.}
The distance from the centre to the external point is: \[ OP = 13 \text{ cm} \] The radius of the circle is: \[ OT = 5 \text{ cm} \] We need to find the tangent length: \[ PT = ? \]

Step 3: Apply the Pythagoras theorem.}
Since triangle $OPT$ is right-angled at $T$, we use: \[ OP^2 = OT^2 + PT^2 \] Substituting the values: \[ 13^2 = 5^2 + PT^2 \] \[ 169 = 25 + PT^2 \] \[ PT^2 = 169 - 25 \] \[ PT^2 = 144 \] \[ PT = 12 \text{ cm} \]

Step 4: Compare with the options and conclude.}

• (A) 12 cm: Correct. This is the value obtained by calculation.
• (B) 13 cm: Incorrect. This is the distance from the centre to point $P$, not the tangent length.
• (C) 15 cm: Incorrect. This does not satisfy the right triangle relation.
• (D) 18 cm: Incorrect. This is greater than the hypotenuse, so it is not possible.

Hence, the length of the tangent from $P$ to the circle is 12 cm.
Final Answer:} 12 cm.