AB is a diameter of radius 4.5 cm circle. P is any point on the circumference of circle such that `PNbotAB`. If length of chord PB is 6 cm . Then BN = ?

Options

A. 5 cm
B. 4 cm
C. 6 cm
D. 4.5 cm

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Circle and Given Information
We are given a circle with a diameter AB and a radius of 4.5 cm. The diameter can be calculated as:
\[ \text{Diameter} = 2 \times \text{Radius} = 2 \times 4.5 \text{ cm} = 9 \text{ cm} \]

### Step 2: Identify Points and Relationships
Point P is on the circumference of the circle, and PN is perpendicular to AB. We also know that the length of chord PB is 6 cm.

### Step 3: Draw the Diagram
Draw the circle with diameter AB. Mark points A and B at the ends of the diameter. Mark point P on the circumference and draw a perpendicular line from P to AB, meeting it at point N.

### Step 4: Identify Triangles
In the triangle ABP, angle APB is a right angle because it is inscribed in a semicircle. Thus, triangle ABP is a right triangle.

### Step 5: Use Similar Triangles
We can use the property of similar triangles. The triangles we will consider are triangle ABP and triangle PBN.

- Angle APB = 90° (as established)
- Angle PBN = 90° (since PN is perpendicular to AB)
- Angle ABP = Angle PBN (common angle)

By AA (Angle-Angle) similarity, triangle ABP is similar to triangle PBN.

### Step 6: Set Up the Proportions
From the similarity of triangles, we can set up the following proportion:
\[ \frac{AB}{PB} = \frac{PB}{BN} \]

### Step 7: Substitute Known Values
We know:
- \( AB = 9 \text{ cm} \)
- \( PB = 6 \text{ cm} \)

Substituting these values into the proportion gives:
\[ \frac{9}{6} = \frac{6}{BN} \]

### Step 8: Cross-Multiply to Solve for BN
Cross-multiplying gives:
\[ 9 \cdot BN = 6 \cdot 6 \]
\[ 9 \cdot BN = 36 \]

### Step 9: Solve for BN
Now, divide both sides by 9:
\[ BN = \frac{36}{9} = 4 \text{ cm} \]

### Final Answer
Thus, the length of BN is **4 cm**.

---