In a circle of radius 7 cm, tangent PT is drawn from a point P such tht PT= 24 cm. If O is the centre of circle, then find the length of OP.

Answer

To find the length of OP in the given circle, we can follow these steps:

### Step 1: Understand the Geometry
In the given problem, we have a circle with center O and radius 7 cm. A tangent PT is drawn from point P outside the circle, and the length of the tangent PT is given as 24 cm. The radius OP will be the hypotenuse of a right triangle formed by the radius OT (which is perpendicular to the tangent PT) and the tangent PT itself.

### Step 2: Identify the Right Triangle
Since the radius (OT) is perpendicular to the tangent (PT) at point T, we can form a right triangle OPT where:
- OT = radius of the circle = 7 cm (one leg of the triangle)
- PT = length of the tangent = 24 cm (the other leg of the triangle)
- OP = length we need to find (the hypotenuse of the triangle)

### Step 3: Apply the Pythagorean Theorem
According to the Pythagorean theorem, in a right triangle:
\[ OP^2 = OT^2 + PT^2 \]

Substituting the known values:
\[ OP^2 = 7^2 + 24^2 \]
\[ OP^2 = 49 + 576 \]
\[ OP^2 = 625 \]

### Step 4: Solve for OP
Now, we take the square root of both sides to find OP:
\[ OP = \sqrt{625} \]
\[ OP = 25 \, \text{cm} \]

### Conclusion
The length of OP is 25 cm.

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