Two tangents PA and PB are drawn from a point P to the circle. If the radius of the circle is 5 cm and AB = 6 cm and O is the centre of the circle. OP cuts AB at C and OC = 4 cm, then OP :-

Options

A. `(25)/(4)` cm
B. 25 cm
C. 13 cm
D. None of these

Answer

To solve the problem, we need to find the length of OP given the radius of the circle, the length of the tangent segment AB, and the length OC. Here are the steps to find OP:

### Step 1: Understand the Geometry
We have a circle with center O and radius 5 cm. Two tangents PA and PB are drawn from point P to points A and B on the circle. The length of the tangent segment AB is given as 6 cm. The line OP intersects AB at point C, and OC is given as 4 cm.

### Step 2: Identify the Right Triangles
Since PA and PB are tangents to the circle, we know that OA and OB are perpendicular to the tangents at points A and B, respectively. Thus, triangles OAP and OBP are right triangles.

### Step 3: Use the Pythagorean Theorem
In triangle OAP, we can apply the Pythagorean theorem:
- OA (radius) = 5 cm
- AC (half of AB) = 6 cm / 2 = 3 cm
- OP is the hypotenuse we need to find.

Using the Pythagorean theorem:
\[ OP^2 = OA^2 + AC^2 \]
\[ OP^2 = 5^2 + 3^2 \]
\[ OP^2 = 25 + 9 \]
\[ OP^2 = 34 \]
\[ OP = \sqrt{34} \]

### Step 4: Calculate the Length of OP
Now we can calculate the value of OP:
\[ OP = \sqrt{34} \approx 5.83 \text{ cm} \]

### Final Answer
Thus, the length of OP is approximately 5.83 cm.

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