(a) The ratio of the interior angle to the exterior angle of a regular polygon is 5 : 2, Find the number of sides of the polygon. (b) The diagram shows a ci...

Answer Details

(a) Number of sides of the polygon

At each vertex the interior and exterior angles are supplementary, so they add up to \(180^\circ\). The ratio interior : exterior is \(5:2\), giving \(5+2 = 7\) equal parts.

\[\text{Exterior angle} = \frac{2}{7}\times 180^\circ = \frac{360^\circ}{7}\]

For any regular polygon the exterior angles sum to \(360^\circ\), so the number of sides is

\[n = \frac{360^\circ}{\text{exterior angle}} = \frac{360^\circ}{\tfrac{360^\circ}{7}} = 7\]

The polygon has 7 sides (a regular heptagon).

(b) Finding \(\angle PRS\)

From the diagram, \(TPQU\) is a straight line through the two points \(P\) and \(Q\) on the circle, and \(PQRS\) is a cyclic quadrilateral. The given angles are \(\angle TPS = 74^\circ\), \(\angle UQR = 68^\circ\) and \(\angle QSR = 40^\circ\).

Step 1: Interior angles of the cyclic quadrilateral. Since \(TPQ\) is a straight line,

\[\angle SPQ = 180^\circ - \angle TPS = 180^\circ - 74^\circ = 106^\circ\]

Since \(PQU\) is a straight line,

\[\angle PQR = 180^\circ - \angle UQR = 180^\circ - 68^\circ = 112^\circ\]

Step 2: Use the opposite-angles property of a cyclic quadrilateral.

\[\angle SRQ = 180^\circ - \angle SPQ = 180^\circ - 106^\circ = 74^\circ\]

Step 3: Split \(\angle SRQ\) using angles in the same segment. The angle \(\angle PRQ\) stands on chord \(PQ\), as does \(\angle PSQ\); angles in the same segment are equal, so \(\angle PRQ = \angle PSQ\).

Now find \(\angle PSQ\). We have \(\angle PSR = 180^\circ - \angle PQR = 180^\circ - 112^\circ = 68^\circ\), and this splits as \(\angle PSR = \angle PSQ + \angle QSR\). Therefore

\[\angle PSQ = 68^\circ - 40^\circ = 28^\circ \;\Rightarrow\; \angle PRQ = 28^\circ\]

Step 4: Obtain \(\angle PRS\). Since \(\angle SRQ = \angle PRS + \angle PRQ\),

\[\angle PRS = 74^\circ - 28^\circ = \boxed{46^\circ}\]

Hence \(\angle PRS = 46^\circ\).