From a point Q, 13 cm away from the centre of a circle, the length of tangent PQ to the circle is 12 cm. What will be it the radius of the circle (in cm)?

Answer

To find the radius of the circle given the distance from point Q to the center O of the circle and the length of the tangent PQ, we can use the Pythagorean theorem. Here’s the step-by-step solution:

### Step 1: Identify the Given Information
- Distance from point Q to the center O of the circle (QO) = 13 cm
- Length of the tangent PQ = 12 cm

### Step 2: Understand the Geometry
In the right triangle formed by points P, Q, and O:
- OQ is the hypotenuse.
- PQ is one leg (the tangent).
- PO is the other leg (the radius of the circle).

### Step 3: Apply the Pythagorean Theorem
According to the Pythagorean theorem:
\[ QO^2 = PQ^2 + PO^2 \]
Substituting the known values:
\[ 13^2 = 12^2 + PO^2 \]

### Step 4: Calculate the Squares
Calculate \(13^2\) and \(12^2\):
\[ 169 = 144 + PO^2 \]

### Step 5: Solve for PO^2
Rearranging the equation to find \(PO^2\):
\[ PO^2 = 169 - 144 \]
\[ PO^2 = 25 \]

### Step 6: Find PO (the Radius)
Taking the square root of both sides to find PO:
\[ PO = \sqrt{25} = 5 \text{ cm} \]

### Conclusion
The radius of the circle is 5 cm.

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