The length of tangent from a point A at a distance of 12 cm from the centre of the circle is 9 cm. What is the radius of the circle ?

Answer

To find the radius of the circle, we can use the Pythagorean theorem. Let's denote:

- O as the center of the circle
- A as the point outside the circle from which the tangent is drawn
- B as the point where the tangent touches the circle

We know the following:

- The distance OA (from the center O to point A) = 12 cm
- The length of the tangent AB = 9 cm
- The radius OB = r (which we need to find)

Since the radius OB is perpendicular to the tangent AB at point B, triangle OAB is a right triangle. According to the Pythagorean theorem:

\[ OA^2 = OB^2 + AB^2 \]

Substituting the known values:

1. \( OA = 12 \, \text{cm} \)
2. \( AB = 9 \, \text{cm} \)

Now, substituting these values into the equation:

\[ 12^2 = OB^2 + 9^2 \]

Calculating the squares:

\[ 144 = OB^2 + 81 \]

Now, isolate \( OB^2 \):

\[ OB^2 = 144 - 81 \]

Calculating the right side:

\[ OB^2 = 63 \]

Taking the square root of both sides to find OB:

\[ OB = \sqrt{63} \]

We can simplify \( \sqrt{63} \):

\[ OB = \sqrt{9 \times 7} = 3\sqrt{7} \, \text{cm} \]

Thus, the radius of the circle is:

\[ \text{Radius} = 3\sqrt{7} \, \text{cm} \]