Mathematics

Prove that the line segments joining the points of contact of two parallel tangents of a circle,passes through its centre ?


SOLUTION
Let, $$PQ$$ and $$RS$$ be two parallel tangents to a circle with cantre $$O$$ where $$M,N$$ be the points of contact of the two tangents $$PQ,RS$$ to the circle respectively.
We have to prove that the line $$MN$$ passes through the centre $$O$$ of the circle.
Here, $$OM$$ and $$OA$$ are joined.
$$OA\parallel PQ\\ PM\parallel AO\\ \therefore \angle PMO+\angle AOM={ 180 }^{ o }\quad \quad \left[ sum\quad of\quad two\quad adjacent\quad interior\quad angles\quad  \right] $$
but, $$\angle PMO=90^{ o }\quad \quad \left[ \because A\quad tangent\quad to\quad a\quad circle\quad is\quad perpendicular\quad to\quad the\quad radius\quad through\quad point\quad of\quad contact \right] $$
So, $$90^{ o }+\angle AOM={ 180 }^{ o }\\ \Rightarrow \angle AOM=90^{ o }\\ \therefore \angle AON=90^{ o }\\ \angle AOM+\angle AON={ 180 }^{ o }$$
$$\therefore$$ $$MN$$ is a straight line and it passes through $$O$$ which is the centre of the given circle.
Hence, proved.
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Subjective Medium Published on 09th 09, 2020
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