Two chords AB and PQ of a circle mutually intersect at an outside point D. If `AD = 12` cm, `AB = 8` cm, `DQ = 6` cm then find `PQ and PD`.

Options

A. A)2&6
B. B)2&8
C. C)3&6
D. D)5&7

Answer

To solve the problem step by step, we will use the properties of intersecting chords in a circle. The formula we will use is:

\[ DB \times DA = DQ \times DP \]

Where:
- \( D \) is the point outside the circle where the two chords intersect.
- \( A \) and \( B \) are points on chord \( AB \).
- \( P \) and \( Q \) are points on chord \( PQ \).

Given:
- \( AD = 12 \) cm
- \( AB = 8 \) cm
- \( DQ = 6 \) cm

We need to find \( PQ \) and \( PD \).

### Step 1: Calculate \( DB \)

Since \( AB \) is a chord and \( AD \) is the distance from point \( D \) to point \( A \), we can find \( DB \) as follows:

\[ DB = AD - AB = 12 \, \text{cm} - 8 \, \text{cm} = 4 \, \text{cm} \]

### Step 2: Set up the equation using the intersecting chords theorem

Using the formula:

\[ DB \times DA = DQ \times DP \]

Substituting the known values:

\[ 4 \, \text{cm} \times 12 \, \text{cm} = 6 \, \text{cm} \times DP \]

### Step 3: Solve for \( DP \)

Calculating the left side:

\[ 48 = 6 \times DP \]

Now, divide both sides by 6 to find \( DP \):

\[ DP = \frac{48}{6} = 8 \, \text{cm} \]

### Step 4: Calculate \( PQ \)

Since \( DP \) is the total length from \( D \) to \( P \) and \( DQ \) is given as 6 cm, we can find \( PQ \) as follows:

\[ PQ = DP - DQ = 8 \, \text{cm} - 6 \, \text{cm} = 2 \, \text{cm} \]

### Final Results

Thus, we have:

- \( PQ = 2 \, \text{cm} \)
- \( PD = 8 \, \text{cm} \)