Two circles of radii 5 and 3 cm, respectively, intersect at two points. At either point of intersection,
the tangent lines to the circles form a 60◦ angle, as in Figure 2.2.4 above. Find the distance
between the centers of the circles
To find the distance between the centers of the circles, we can use the Pythagorean Theorem.
Let's label the centers of the circles A and B.
Let the distance between the centers be x.
Since the tangents at the points of intersection form a 60° angle, we can draw a triangle with sides A, B, and x, where A and B are the radii of the circles.
Now, we can use the Law of Cosines to find x.
The Law of Cosines states that in a triangle with sides a, b, and c and angle C opposite to side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
Plugging in the values, we have:
x^2 = 5^2 + 3^2 - 2 * 5 * 3 * cos(60°)
Simplifying further,
x^2 = 25 + 9 - 30 * cos(60°)
Since cos(60°) = 1/2, we have:
x^2 = 25 + 9 - 30 * (1/2)
x^2 = 25 + 9 - 15
x^2 = 19
To solve for x, we take the square root of both sides:
x = √19
Therefore, the distance between the centers of the circles is √19 cm.