Question

Two circles of radii 6 cm and 6 cm intersect each other. The distance between their centers is 8 cm. What is the length of their common chord?

• A. \(4\sqrt{5}\) cm
• B. \(6\sqrt{2}\) cm
• C. 8 cm
• D. 6 cm

Hint:

If a geometry problem's given numbers don't lead to any of the multiple-choice answers, check for a simple typo that might make the problem symmetric or easier. Assuming equal radii is a common correction.

Answer 1

The Correct Option is A

Approach: The line joining two centres is the perpendicular bisector of the common chord. With equal radii, that line is cut in half, so one right triangle and Pythagoras finishes it.

Step 1: Set up the picture.
Centres \(C_1, C_2\) with \(r_1 = r_2 = 6\) cm, and \(C_1C_2 = 8\) cm. Let the common chord meet \(C_1C_2\) at \(M\); then \(C_1C_2 \perp\) chord and \(M\) bisects the chord.

Step 2: Locate the foot M.
Because both radii are equal, the figure is symmetric, so \(M\) is the midpoint of \(C_1C_2\):
\[ C_1M = \frac{8}{2} = 4 \text{ cm}. \]

Step 3: Pythagoras in triangle \(C_1MA\).
Let \(AM = h\) be half the chord. Then
\[ h^2 + 4^2 = 6^2 \implies h^2 = 36 - 16 = 20 \implies h = 2\sqrt{5}. \]

Step 4: Double it for the full chord.
\[ \text{Chord} = 2h = 2 \times 2\sqrt{5} = 4\sqrt{5} \text{ cm}. \]

Final Answer: \(\boxed{4\sqrt{5}\text{ cm}}\). (Option 1)

Answer 2

Approach (area method): The triangle formed by the two centres and one intersection point has known sides 6, 6, 8. Its area, found two ways, gives the perpendicular height to the chord, which is half the chord.

Step 1: Semi-perimeter \( s=\frac{6+6+8}{2}=10 \). By Heron's formula:
\[ \text{Area} = \sqrt{10(10-6)(10-6)(10-8)} = \sqrt{10\times4\times4\times2} = \sqrt{320} = 8\sqrt{5}. \]
Step 2: Taking the side of length 8 (the line joining centres) as base, the height \( h \) from the intersection point satisfies \( \text{Area} = \frac{1}{2}\times8\times h \):
\[ 8\sqrt{5} = 4h \implies h = 2\sqrt{5}. \]
Step 3: This height is exactly half the common chord (since the line of centres bisects it perpendicularly), so:
\[ \text{Chord} = 2h = 4\sqrt{5}\text{ cm}. \]

Final Answer: \( \boxed{4\sqrt{5}\text{ cm}} \). (Option 1)