The diameter of a circle with centre at C is 50 cm. CP is a radial segment of the circle. AB is a chord perpendicular to CP and passes through P. CP produced intersects the circle at D. If DP = 18 cm, then what is the length of AB?

Options

A. 24 cm
B. 32 cm
C. 40 cm
D. 48 cm

Answer

To solve the problem step by step, we will follow the geometric properties of circles and chords.

### Step 1: Identify the given information
- The diameter of the circle is 50 cm, which means the radius (r) is half of the diameter:
\[ r = \frac{50}{2} = 25 \text{ cm} \]
- CP is a radial segment, and DP is given as 18 cm.

### Step 2: Calculate the length of CP
Since D is on the circle and CP is a radial segment, we can find the length of CP using the relationship:
\[ CD = CP + PD \]
Given that DP = 18 cm, we can express CD as:
\[ CD = r = 25 \text{ cm} \]
Thus,
\[ CP + 18 = 25 \]
Solving for CP:
\[ CP = 25 - 18 = 7 \text{ cm} \]

### Step 3: Use the property of perpendicular chords
Since AB is a chord perpendicular to CP at point P, we can use the property that states if a radius (or diameter) is perpendicular to a chord, it bisects the chord. Let AP = PB = x.

### Step 4: Apply the intersecting chords theorem
According to the intersecting chords theorem:
\[ AP \cdot PB = CP \cdot PD \]
Substituting the values we have:
\[ x \cdot x = CP \cdot PD \]
\[ x^2 = 7 \cdot 18 \]
Calculating the right side:
\[ x^2 = 126 \]
Taking the square root:
\[ x = \sqrt{126} = \sqrt{9 \cdot 14} = 3\sqrt{14} \text{ cm} \]

### Step 5: Calculate the length of chord AB
Since AB = AP + PB = x + x = 2x:
\[ AB = 2x = 2(3\sqrt{14}) = 6\sqrt{14} \text{ cm} \]

### Step 6: Approximate the length of AB
To find the numerical value of AB, we can approximate \(\sqrt{14}\):
\[ \sqrt{14} \approx 3.74 \Rightarrow AB \approx 6 \cdot 3.74 = 22.44 \text{ cm} \]

### Conclusion
Thus, the length of chord AB is approximately \(22.44 \text{ cm}\).

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