Two equal circles of radius r intersect such that each passes through the centre of the other. Find the length of the common chord.
To find the length of the common chord, we can start by drawing a diagram to visualize the problem.
Let's assume we have two circles, labeled as Circle 1 and Circle 2, with equal radii r. Both circles intersect each other such that each passes through the center of the other. This means that the point where the circles intersect is also the center of each circle.
Drawing a line connecting the two intersection points will give us the common chord. Let's label this line as the common chord.
Now, let's analyze the diagram to find the length of the common chord.
Since the two circles have equal radii, the line connecting the center of each circle to the point of intersection will be perpendicular to the common chord. Let's label the intersection point as P, the center of Circle 1 as O1, and the center of Circle 2 as O2.
We can observe that triangle O1PO2 is an isosceles right triangle, with the hypotenuse O1O2 being the radius r of each circle.
Since triangle O1PO2 is isosceles, the two legs O1P and O2P are of equal length, which is also equal to r.
Let's label the length of the common chord as C.
Now, we can divide the common chord C into two equal segments, each connecting an endpoint of C to the midpoint of O1O2.
Notice that this divides the common chord C into two right triangles with hypotenuses O1P and O2P, and with legs equal to half the length of the common chord.
Using the Pythagorean theorem, we can write the equation:
(C/2)^2 + r^2 = r^2
Simplifying this equation, we get:
C^2/4 = 0
Multiplying both sides of the equation by 4, we have:
C^2 = 0
Taking the square root of both sides of the equation, we get:
C = 0
Therefore, the length of the common chord is 0.