Question

In the figure, the tangents $ PQ$ and $ PR$ are drawn to a circle such that $ \angle RPQ = 30°$. A chord $ RS$ is drawn parallel to the tangent $ PQ$. Find the $ \angle RQS$.

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Answer 1

Step 1: State the given data and equate $\angle PQR$ & $\angle PRQ$

As given, $PQ$ and $PR$ are the tangents to the circle.

$\angle RPQ=30°$

And, $RS$ is a chord such that $RS\left|\right|PQ$.

Now, since $PQ$ and $PR$ are the tangents drawn from the same point.

So, $PQ=PR$

In $△PQR$, using the property of isosceles triangles,

$\because$ $PQ=PR$

$\therefore$ $\angle PQR=\angle PRQ$ $...\left(\mathrm{i}\right)$

Step 2: Calculate the value of $\angle PQR$ and $\angle PRQ$

Now, Using the angle sum property in the $△PQR$,

$\Rightarrow$ $\angle PQR+\angle PRQ+\angle RPQ=180°$

$\Rightarrow$ $\angle PQR+\angle PQR+\angle RPQ=180°$ [Using equation $\left(\mathrm{i}\right)$]

$\Rightarrow$ $\angle PQR+\angle PQR+30°=180°$ $\left[\because \angle RPQ=30°\right]$

$\Rightarrow$ $2\angle PQR+30°=180°$

$\Rightarrow$ $2\angle PQR=180°-30°$

$\Rightarrow$ $2\angle PQR=150°$

$\Rightarrow$ $\angle PQR=\frac{150°}{2}$

$\Rightarrow$ $\angle PQR=75°$

So, $\angle PRQ=\angle PQR$

$\Rightarrow$ $\angle PRQ=75°$

Step 3: Calculate the value of $\angle SRQ$ and $\angle QSR$

Since $RS\left|\right|PQ$ and $QR$ is a transversal.

So, $\angle SRQ=\angle PQR$ [alternate interior angles]

$\Rightarrow$ $\angle SRQ=75°$

Now, according to the alternate segment theorem, the angle between a tangent and a chord of a circle is always equals to the angle made by the chord in the alternate segment of the circle.

So, $\angle QRP=\angle QSR$

Or, $\angle QSR=\angle QRP$

$\Rightarrow$ $\angle QSR=75°$ $\left[\because \angle QRP=\angle PRQ=75°\right]$

Step 4: Calculate the value of $\angle RQS$

Applying the angle sum property in the $\angle QRS$,

$\angle SRQ+\angle RQS+\angle QSR=180°$

$\Rightarrow$ $75°+\angle RQS+75°=180°$

$\Rightarrow$ $150°+\angle RQS=180°$

$\Rightarrow$ $\angle RQS=180°-150°$

$\Rightarrow$ $\angle RQS=30°$

Hence, the value of $\angle RQS$ is $30°$.