From a point P, two tangents PA and PB are drawn to a circle C(O,r). If OP=2r, show that △APB is an equilateral triangle.
Answer:
- Let OP meet the circle at Q. Join OA and AQ.
- We know that the radius through the point of contact is perpendicular to the tangent.
- The circle is represented as , this means that is the center of the circle and is its radius.
Also, we see that
Substituting the value of and in the above equation, we have - As, is the mid-point of is the median from the vertex to the hypotenuse of the right-angled triangle .
We know that the median on the hypotenuse of a right- angled triangle is half of its hypotenuse.
Thus, - We know that the sum of angles of a triangle is
For , Also, two tangents from an external point are equally inclined to the line segment joining the center to that point.
So, - The lengths of the tangents drawn from an external point to a circle are equal.
So, - Consider Similarly,
- As all the angles of the measure , it is an triangle.