Two concentric circles with centre P have radii 6.5 cm and 3.3 cm. Through a point A of the larger circle, a tangent is drawn to the smaller circle touching it at B and the larger circle at C. find the length of the tangent

Options

A. 5.6 cm
B. 11.2 cm
C. 11.8 cm
D. 6.5 cm

Answer

To find the length of the tangent drawn from point A on the larger circle to the smaller circle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Radius of the larger circle (R) = 6.5 cm
- Radius of the smaller circle (r) = 3.3 cm
- Let point A be on the circumference of the larger circle, and point B be the point where the tangent touches the smaller circle.

2. **Draw the Diagram:**
- Draw two concentric circles with center P.
- Mark point A on the circumference of the larger circle.
- Draw a tangent line from point A that touches the smaller circle at point B.

3. **Understand the Geometry:**
- The line segment PB (from the center P to point B) is the radius of the smaller circle and is perpendicular to the tangent line AC at point B.
- The line segment PA (from the center P to point A) is the radius of the larger circle.

4. **Apply the Pythagorean Theorem:**
- In triangle PAB, we can apply the Pythagorean theorem:
\[ PA^2 = PB^2 + AB^2 \]
- Here, PA is the hypotenuse, PB is one leg, and AB is the other leg.

5. **Substitute the Known Values:**
- Substitute the values of PA and PB:
\[ (6.5)^2 = (3.3)^2 + AB^2 \]

6. **Calculate the Squares:**
- Calculate \( PA^2 \) and \( PB^2 \):
\[ 6.5^2 = 42.25 \]
\[ 3.3^2 = 10.89 \]

7. **Set Up the Equation:**
- Substitute the squared values into the equation:
\[ 42.25 = 10.89 + AB^2 \]

8. **Solve for AB^2:**
- Rearranging gives:
\[ AB^2 = 42.25 - 10.89 = 31.36 \]

9. **Take the Square Root:**
- Find AB by taking the square root:
\[ AB = \sqrt{31.36} = 5.6 \text{ cm} \]

10. **Calculate the Length of the Tangent AC:**
- Since AC is twice the length of AB (because PB bisects AC):
\[ AC = 2 \times AB = 2 \times 5.6 = 11.2 \text{ cm} \]

### Final Answer:
The length of the tangent AC is **11.2 cm**.