Two chords AB and CD of a circle intersect each other at P internally. If AP = 3.5 cm, PC = 5 cm, and DP = 7 cm, then what is the measure of PB?

Options

A. 8 cm
B. 12 cm
C. 10 cm
D. 10.5 cm

Answer

To solve the problem step by step, we can use the property of intersecting chords in a circle. The property states that if two chords intersect each other inside a circle, then the products of the lengths of the segments of each chord are equal.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Let \( AP = 3.5 \, \text{cm} \)
- Let \( PC = 5 \, \text{cm} \)
- Let \( DP = 7 \, \text{cm} \)
- We need to find \( PB \).

2. **Apply the Intersecting Chords Theorem:**
According to the theorem:
\[ AP \times PB = PC \times PD \]

3. **Substitute the Known Values:**
Substitute the known lengths into the equation:
\[ 3.5 \times PB = 5 \times 7 \]

4. **Calculate the Right Side:**
Calculate \( 5 \times 7 \):
\[ 5 \times 7 = 35 \]
So the equation becomes:
\[ 3.5 \times PB = 35 \]

5. **Solve for \( PB \):**
To find \( PB \), divide both sides by \( 3.5 \):
\[ PB = \frac{35}{3.5} \]

6. **Simplify the Division:**
To simplify \( \frac{35}{3.5} \), we can multiply the numerator and denominator by 10 to eliminate the decimal:
\[ PB = \frac{35 \times 10}{3.5 \times 10} = \frac{350}{35} = 10 \]

7. **Final Answer:**
Thus, the length of \( PB \) is:
\[ PB = 10 \, \text{cm} \]