# 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.