If the angle between two tangents drawn from an external point P to a circle of radius r and center O is 60°, then find the length of OP.

We are given a circle with center O and radius r, and an external point P from which two tangents are drawn to the circle. The angle between these tangents is 60°. We need to find the length of OP, where OP is the distance from point P to the center O of the circle.

Step 1: Use the properties of tangents
Let the points of contact of the tangents with the circle be A and B. The tangents from an external point to a circle are equal in length. So, the lengths of the tangents PA and PB are equal.

Also, the radius of the circle is perpendicular to the tangent at the point of contact. Therefore, the lines OA and OB are perpendicular to the tangents PA and PB, respectively.

Step 2: Consider the triangle OAP and OBP
Since OA and OB are radii of the circle, they are equal in length. Therefore, triangle OAP and triangle OBP are congruent.

Let the angle between the two tangents be 60°. This means that the angle ∠APB between the two tangents is 60°.

Step 3: Use the formula for the length of OP
From the geometry of the situation, the following relation holds for the length of OP (denoted as OP):

OP = √(OA² + AB²)

Now, since the angle between the tangents is 60°, the angle ∠APB can be used to calculate the length of OP.

By using the formula for the distance from an external point to the center of the circle:

OP = √(r² + r² / sin²(θ/2))

Where θ = 60° and r is the radius of the circle.

Step 4: Calculate OP
Substituting θ = 60° into the formula:

OP = √(r² + r² / sin²(30°))

Since sin(30°) = 1/2:

OP = √(r² + r² / (1/4))

OP = √(r² + 4r²)

OP = √(5r²)

OP = r√5

Final Answer:
The length of OP is r√5.