Final answer:
By creating a right triangle within the intersecting circles, applying the Pythagorean theorem, and simplifying the equation, we find that the length of the common chord is √3 * r (option C).
Step-by-step explanation:
The question asks to find the length of a common chord in two equal circles, where each circle passes through the center of the other. This particular type of problem is linked with the concept of intersecting circles in geometry. To solve this, we can apply the Pythagorean theorem to the right triangle formed by the radius, half of the chord, and the distance from the center to the midpoint of the chord.
Let's denote the length of the common chord as L. Since the radius r of each circle extends to the center of the other, creating an equilateral triangle within the intersecting portion, we can say that L/2 (half the chord length) and the radius r form a right triangle with the segment connecting the center of one circle to the midpoint of the chord (which is also r in length). Using the Pythagorean theorem, we get:
(L/2)^2 + r^2 = (2r)^2
Simplifying this equation, we get L = √3 * r, which corresponds to the option C.