From a point Q, the length of the tangent to a circle is 12 cm and distance of d from the centre is 13 cm. Find the radius of the circle.

Answer

To find the radius of the circle given the length of the tangent and the distance from the point to the center of the circle, we can follow these steps:

### Step 1: Understand the Given Information
We are given:
- Length of the tangent (QT) = 12 cm
- Distance from point Q to the center of the circle (OQ) = 13 cm

### Step 2: Set Up the Right Triangle
From point Q, a tangent QT is drawn to the circle. The radius OT is perpendicular to the tangent at point T. Therefore, we can form a right triangle OQT, where:
- OQ is the hypotenuse,
- OT is one leg (the radius we need to find),
- QT is the other leg (the length of the tangent).

### Step 3: Apply the Pythagorean Theorem
According to the Pythagorean theorem:
\[ OQ^2 = OT^2 + QT^2 \]
Where:
- OQ = 13 cm (the distance from Q to the center O)
- QT = 12 cm (the length of the tangent)

### Step 4: Substitute the Values
Substituting the known values into the equation:
\[ 13^2 = OT^2 + 12^2 \]

Calculating the squares:
\[ 169 = OT^2 + 144 \]

### Step 5: Solve for OT^2
Now, isolate OT^2:
\[ OT^2 = 169 - 144 \]
\[ OT^2 = 25 \]

### Step 6: Take the Square Root
To find OT (the radius of the circle), take the square root of both sides:
\[ OT = \sqrt{25} = 5 \text{ cm} \]

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

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