A point Q is 13 cm from the centre of a circle . The lenght of the tangent drawn from Q to a circle is 12 cm. The distance of Q from the nearest point of the circle is

Options

A. 7 cm
B. 8 cm
C. 5 cm
D. 12 cm

Answer

To solve the problem step by step, we will use the properties of circles and the Pythagorean theorem.

### Step 1: Understand the problem
We have a point Q that is 13 cm away from the center O of a circle. A tangent drawn from point Q to the circle has a length of 12 cm. We need to find the distance from point Q to the nearest point on the circle.

### Step 2: Identify the relevant elements
- Let O be the center of the circle.
- Let P be the point where the tangent touches the circle.
- The distance from O to Q is given as 13 cm (OQ = 13 cm).
- The length of the tangent (QP) is given as 12 cm.

### Step 3: Use the Pythagorean theorem
In the right triangle OQP:
- OQ is the hypotenuse (13 cm).
- QP is one leg (12 cm).
- OP is the radius of the circle (let's denote it as R).

According to the Pythagorean theorem:
\[ OQ^2 = OP^2 + QP^2 \]

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

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

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

Now, take the square root to find R:
\[ R = \sqrt{25} = 5 \text{ cm} \]

### Step 6: Find the distance from Q to the nearest point on the circle
The nearest point on the circle from point Q is along the line connecting Q to O. The distance from Q to the nearest point on the circle is given by:
\[ \text{Distance} = OQ - R \]
\[ \text{Distance} = 13 \text{ cm} - 5 \text{ cm} = 8 \text{ cm} \]

### Conclusion
The distance of Q from the nearest point of the circle is **8 cm**.

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