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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Subjects 10
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