The two chords AB and CD of a circle intersect at point P, such that BP=4 cm, PD=5 cm, and CP=8 cm. find the length of chord AB.

Options

A. 8
B. 10
C. 12
D. 14

Answer

To find the length of chord AB in the circle where the two chords AB and CD intersect at point P, we can use the property of intersecting chords. The property states that if two chords intersect each other, the products of the lengths of the segments of each chord are equal.

Let's denote:
- BP = 4 cm
- PD = 5 cm
- CP = 8 cm
- AP = x cm (this is what we need to find)

According to the intersecting chord theorem:
\[ AP \times BP = CP \times PD \]

### Step 1: Write the equation based on the intersecting chords theorem
Using the values we have:
\[ x \times 4 = 8 \times 5 \]

### Step 2: Calculate the right side of the equation
Calculating the right side:
\[ 8 \times 5 = 40 \]
So, we have:
\[ 4x = 40 \]

### Step 3: Solve for x
Now, divide both sides by 4:
\[ x = \frac{40}{4} = 10 \]
Thus, AP = 10 cm.

### Step 4: Find the length of chord AB
Now that we have AP, we can find the length of chord AB:
\[ AB = AP + PB \]
Substituting the known values:
\[ AB = 10 + 4 = 14 \text{ cm} \]

### Final Answer
The length of chord AB is **14 cm**.

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