Point A is situated at a distance of 6.5 cm from the centre of a circle. The length of tangent drawn from point A to the circle is 6 cm. What is the radius of the circle?

Options

A. A)5 cm
B. B)4 cm
C. C)3.5 cm
D. D)2.5 cm

Answer

To solve the problem step by step, we will use the information given about the circle, the point A, and the tangent drawn from point A to the circle.

### Step 1: Understand the Given Information
- Distance from the center of the circle (O) to point A = 6.5 cm
- Length of the tangent from point A to the circle (AT) = 6 cm
- We need to find the radius of the circle (r).

### Step 2: Draw the Diagram
Draw a circle with center O. Mark point A outside the circle such that OA = 6.5 cm. Draw a tangent from point A to the circle, touching the circle at point T. The length of the tangent AT = 6 cm.

### Step 3: Apply the Tangent-Secant Theorem
According to the Tangent-Secant Theorem:
\[ AT^2 = AP \times AQ \]
Where:
- AT = length of the tangent = 6 cm
- AP = distance from point A to the point where the secant intersects the circle (which we will find)
- AQ = distance from point A to the other point on the circle (which we will find)

### Step 4: Express AP and AQ in Terms of r
Let r be the radius of the circle. Then:
- AP = OA - OT = 6.5 - r
- AQ = OA + OT = 6.5 + r

### Step 5: Substitute Values into the Equation
Substituting the values into the Tangent-Secant Theorem:
\[ 6^2 = (6.5 - r)(6.5 + r) \]
\[ 36 = (6.5)^2 - r^2 \]
\[ 36 = 42.25 - r^2 \]

### Step 6: Solve for r^2
Rearranging the equation gives:
\[ r^2 = 42.25 - 36 \]
\[ r^2 = 6.25 \]

### Step 7: Find the Value of r
Taking the square root of both sides:
\[ r = \sqrt{6.25} \]
\[ r = 2.5 \, \text{cm} \]

### Final Answer
The radius of the circle is **2.5 cm**.

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