Question

A 5 cm long perpendicular is drawn from the centre of a circle to a 24 cm long chord. Find the diameter of the circle.

• A. 26 cm
• B. 32 cm
• C. 13 cm
• D. 30 cm

Hint:

Remember the standard Pythagorean triple $(5, 12, 13)$.
When you see a perpendicular of $5$ and a half-chord of $12$, the radius must be $13$.
Double this value immediately to get the diameter: \( 13 \times 2 = 26\text{ cm} \).
This saves valuable time during competitive exams.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
In this problem, we are given a circle with a chord of known length and the perpendicular distance from the center of the circle to this chord.
We need to determine the diameter of the circle.
This is a standard geometry problem that involves the relationship between the radius, the chord, and the perpendicular distance.
We will use circle geometry theorems and the Pythagorean theorem to find the radius and then compute the diameter.

Step 2: Key Formula or Approach:

• A perpendicular line segment drawn from the center of a circle to any of its chords bisects the chord into two equal parts.
  
• The radius ($r$), the perpendicular distance ($p$), and half of the chord length ($c/2$) form a right-angled triangle.
  
• The Pythagorean theorem states: \( r^2 = p^2 + (c/2)^2 \), where $r$ is the radius, $p$ is the perpendicular distance, and $c$ is the chord length.
  
• The diameter ($d$) of the circle is twice its radius: \( d = 2r \).
  

Step 3: Detailed Explanation:

• Let the center of the circle be represented by $O$.
  
• Let the chord be represented by $AB$, with a given length of $24\text{ cm}$.
  
• Let the perpendicular line segment from $O$ to the chord $AB$ meet the chord at point $M$.
  
• The length of this perpendicular distance $OM$ is given as $5\text{ cm}$.
  
• According to the property of circles, the perpendicular from the center bisects the chord. Thus, point $M$ is the midpoint of $AB$.
  
• This gives the length of the half-chord: \( AM = MB = \frac{AB}{2} = \frac{24}{2} = 12\text{ cm} \).
  
• Now, we can construct a right-angled triangle $\triangle OMA$, where the hypotenuse $OA$ represents the radius $r$ of the circle.
  
• Applying the Pythagorean theorem to $\triangle OMA$:
  \[ OA^2 = OM^2 + AM^2 \]
  
• Substituting the given numerical values into the equation:
  \[ r^2 = 5^2 + 12^2 \]
  
• Evaluating the squares:
  \[ r^2 = 25 + 144 \]
  \[ r^2 = 169 \]
  
• Taking the square root on both sides to find the radius $r$:
  \[ r = \sqrt{169} = 13\text{ cm} \]
  
• To find the diameter $d$ of the circle, we multiply the radius by 2:
  \[ d = 2 \times r = 2 \times 13 = 26\text{ cm} \]
  
• Therefore, the diameter of the circle is $26\text{ cm}$.
  

Step 4: Final Answer:
The diameter of the circle is calculated to be $26\text{ cm}$, which corresponds directly to Option (A).