Question

In a circle with center O, a 6cm long chord is at a distance 4 cm from the center. Then the length of diameter is

• A. 5 cm
• B. 10 cm
• C. 15 cm
• D. 8 cm

Hint:

This problem uses the very common 3-4-5 Pythagorean triple. Whenever you see a right-angled triangle with two sides as 3 and 4 in a geometry problem, the hypotenuse is almost always 5. Recognizing this can save you calculation time. Also, be careful to read the question fully; it asks for the diameter, not the radius, which is a common mistake.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given the length of a chord and its perpendicular distance from the center of a circle. We need to find the diameter of the circle.

Step 2: Key Formula or Approach:
The key geometric property is that the perpendicular from the center of a circle to a chord bisects the chord. This forms a right-angled triangle with the radius of the circle as the hypotenuse, the perpendicular distance from the center as one leg, and half the length of the chord as the other leg. We can use the Pythagorean theorem: \((\text{radius})^2 = (\text{distance})^2 + (\text{half of chord})^2\).

Step 3: Detailed Explanation:
Let \(r\) be the radius of the circle.
Length of the chord, \(L = 6\) cm.
Distance from the center to the chord, \(d = 4\) cm.
The perpendicular from the center bisects the chord. So, the length of half the chord is \(\frac{L}{2} = \frac{6}{2} = 3\) cm.
Now, we have a right-angled triangle with sides:
Hypotenuse = \(r\)
One leg = \(d = 4\) cm
Other leg = \(\frac{L}{2} = 3\) cm
Using the Pythagorean theorem (\(a^2 + b^2 = c^2\)):
\[ d^2 + \left(\frac{L}{2}\right)^2 = r^2 \]
\[ 4^2 + 3^2 = r^2 \]
\[ 16 + 9 = r^2 \]
\[ 25 = r^2 \]
\[ r = \sqrt{25} = 5 \text{ cm} \]
The radius of the circle is 5 cm.
The question asks for the length of the diameter.
Diameter \(D = 2 \times r\).
\[ D = 2 \times 5 = 10 \text{ cm} \]

Step 4: Final Answer:
The length of the diameter is 10 cm.