The angle between two tangents drawn from an external point p to a circle O is 60 ,then find the length of OP?

Question

Answer

Angle between two tangents to a circle

Let's consider a circle O with center O and radius r, and an external point P. Two tangents are drawn from point P to circle O, and the angle between these tangents is given as 60 degrees. We need to find the length of OP, the distance between point P and the center of the circle.

Properties of Tangents

Before we start solving the problem, let's review some important properties of tangents to a circle:

Tangents drawn from an external point to a circle are equal in length.

The radius drawn to the point of tangency is perpendicular to the tangent line.

The angle between a tangent and the radius drawn to the point of tangency is 90 degrees.

Solving the Problem

Based on the properties mentioned above, we can construct the following diagram:

```
P
|\
| \
| \
| \
| \
| \
|______\
O r Q
```

In the diagram, the lines OP and OQ represent the radii of the circle, and the lines PQ and PR represent the tangents drawn from point P to circle O.

Since the tangents drawn from an external point to a circle are equal in length, we can label PQ and PR as equal, let's say x.

Now, we can construct a right-angled triangle OPQ, where angle OPQ is 90 degrees (property 3 mentioned above).

Since the angle between the tangents is given as 60 degrees, angle OPR is also 60 degrees (as tangents are perpendicular to radii).

Using the property of a triangle that the sum of angles in a triangle is 180 degrees, we can find angle OPQ:

```
angle OPQ + angle OPQ + angle OPR = 180 degrees
90 degrees + 90 degrees + 60 degrees = 180 degrees
```

Therefore, angle OPQ is 30 degrees.

Now, we have a right-angled triangle OPQ with angle OPQ equal to 30 degrees and PQ equal to x (the length of the tangents).

We can use trigonometry to find the length of OP. Since OP is the hypotenuse of triangle OPQ, we can use the sine function:

```
sin(30 degrees) = PQ / OP
1/2 = x / OP
OP = 2 * x
```

Therefore, the length of OP is 2 times the length of the tangents.

Conclusion

In this problem, we found that the length of OP, the distance between an external point P and the center O of a circle, is equal to twice the length of the tangents drawn from point P to circle O. We used the properties of tangents and trigonometry to solve the problem and arrived at the solution OP = 2 *

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