A chord AB of length 16 cm is drawn in a circle of diameter 20 cm. If tangents at A and B intersect each other at point P, find the length of PA.

Options

A. `40/3 cm`
B. `20/3 cm`
C. `12 cm`
D. `50/3 cm`

Answer

To solve the problem, we need to find the length of PA where tangents at points A and B intersect at point P, given that the chord AB has a length of 16 cm and the diameter of the circle is 20 cm.

### Step-by-Step Solution:

1. **Identify the Circle's Radius**:
- The diameter of the circle is given as 20 cm.
- Therefore, the radius \( r \) of the circle is:
\[ r = \frac{20}{2} = 10 \text{ cm} \]

**Hint**: Remember that the radius is half of the diameter.

2. **Find the Length of Half the Chord**:
- The length of the chord AB is 16 cm.
- The midpoint M of the chord divides it into two equal segments, so:
\[ AM = MB = \frac{16}{2} = 8 \text{ cm} \]

**Hint**: The midpoint of a chord divides it into two equal parts.

3. **Draw the Right Triangle**:
- Draw a line from the center of the circle C to the midpoint M of the chord AB. This line CM is perpendicular to the chord AB.
- In triangle CAM, we have:
- \( CA = r = 10 \text{ cm} \)
- \( AM = 8 \text{ cm} \)

4. **Use the Pythagorean Theorem**:
- In right triangle CAM, we can apply the Pythagorean theorem:
\[ CM^2 + AM^2 = CA^2 \]
\[ CM^2 + 8^2 = 10^2 \]
\[ CM^2 + 64 = 100 \]
\[ CM^2 = 100 - 64 = 36 \]
\[ CM = \sqrt{36} = 6 \text{ cm} \]

**Hint**: The Pythagorean theorem relates the sides of a right triangle.

5. **Find the Length of Tangent PA**:
- In triangle CAP, we know that:
- \( CA = 10 \text{ cm} \)
- \( CM = 6 \text{ cm} \)
- Let \( PA = L \) (the length of the tangent).
- Again using the Pythagorean theorem in triangle CAP:
\[ CP^2 = CA^2 + AP^2 \]
\[ CP^2 = 10^2 + L^2 \]
- Since \( CP = CM + PA = 6 + L \):
\[ (6 + L)^2 = 100 + L^2 \]
\[ 36 + 12L + L^2 = 100 + L^2 \]
\[ 12L = 100 - 36 \]
\[ 12L = 64 \]
\[ L = \frac{64}{12} = \frac{16}{3} \text{ cm} \]

**Hint**: When using the Pythagorean theorem, ensure to express all sides correctly.

6. **Final Result**:
- The length of PA is:
\[ PA = \frac{16}{3} \text{ cm} \approx 5.33 \text{ cm} \]

### Summary:
The length of PA, where the tangents at points A and B intersect at point P, is \( \frac{16}{3} \) cm.