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 will use the Pythagorean theorem. Here’s the step-by-step solution:

### Step 1: Understand the Geometry
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 OT is perpendicular to the tangent PT at point T.

### Step 2: Identify the Right Triangle
Since OT is perpendicular to PT, we can form a right triangle OPT, where:
- OP is the hypotenuse,
- OT is one leg (the radius of the circle),
- PT is the other leg (the length of the tangent).

### Step 3: Assign Known Values
From the problem:
- OT (radius) = 7 cm
- PT (tangent) = 24 cm

### Step 4: Apply the Pythagorean Theorem
According to the Pythagorean theorem:
\[ OP^2 = OT^2 + PT^2 \]

### Step 5: Substitute the Known Values
Substituting the known values into the equation:
\[ OP^2 = 7^2 + 24^2 \]
\[ OP^2 = 49 + 576 \]

### Step 6: Calculate the Sum
Now, calculate the sum:
\[ OP^2 = 625 \]

### Step 7: Find OP
To find OP, take the square root of both sides:
\[ OP = \sqrt{625} \]
\[ OP = 25 \text{ cm} \]

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

---