From a point PPP, two tangents PAPAPA and PBPBPB are drawn to a circle C(O,r)C(O,r)C(O,r). If OP=2rOP=2rOP=2r, show that APBAPBAPB is an equilateral triangle.


Answer:


Step by Step Explanation:
  1. Let OPOPOP meet the circle at QQQ. Join OAOAOA and AQAQAQ.
    A O Q P B
  2. We know that the radius through the point of contact is perpendicular to the tangent. [Math Processing Error]
  3. The circle is represented as C(O,r)C(O,r), this means that OO is the center of the circle and rr is its radius. [Math Processing Error] Also, we see that OP=OQ+QP.OP=OQ+QP.

    Substituting the value of OPOP and OQOQ in the above equation, we have [Math Processing Error]
  4. As, QQ is the mid-point of OP,AQOP,AQ is the median from the vertex AA to the hypotenuse OPOP of the right-angled triangle AOQAOQ.

    We know that the median on the hypotenuse of a right- angled triangle is half of its hypotenuse.
    Thus, QA=12OP=12(2r)=r.QA=12OP=12(2r)=r. [Math Processing Error]
  5. We know that the sum of angles of a triangle is 180.180.

    For AOPAOP, [Math Processing Error] Also, two tangents from an external point are equally inclined to the line segment joining the center to that point.
    So, [Math Processing Error]
  6. The lengths of the tangents drawn from an external point to a circle are equal.
    So, [Math Processing Error]
  7. Consider PABPAB [Math Processing Error] Similarly, PBA=60.PBA=60.
  8. As all the angles of the PABPAB measure 6060, it is an equilateralequilateral triangle.

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