POR is the diameter of a circle. PQ and RQ are its two chords. The length of arc $$\overline{PQ}$$ is half of the length of are $$\overline{RQ}$$. Find the measure of ∠POQ.​

Correct option is D

Given:

$$\overline{POR}$$​ is the diameter

Arcs: PQ is half the length of arc RQ

Need to find ∠POQ

Concept Used:

The angle subtended by an arc at the circle is directly proportional to the arc’s length. 

Triangle made by diameter to the circle is right angle triangle 

Solution: ​

By the property of diameter of semicircle 

​$$\angle PQR = 90\degree$$ 

arc PQ is half the length of arc RQ

so, arc PQ : arc RQ = 1 : 2  

​$$\angle QRP = x, \quad \angle QPR = 2x$$​​

thus, In $$\triangle PQR$$ 

$$\angle QPR + \angle QRP +\angle PQR = 180 \degree \\ \ \\ 2x+x +90\degree = 180\degree \\ \ \\ 3x = 90\degree \\ \ \\ x = 30\degree$$ 

Now,  

​$$\angle QRP = 30\degree , \quad \angle QPR = 60\degree$$ 

Now, PO = QO  = OR (radius) 

​$$\angle OPQ = \angle OQP = 60\degree$$(equal side make equal angle)

Now, In $$\triangle POQ$$ 

$$\angle OPQ + \angle PQO + \angle POQ = 180\degree \\ \ \\ 60 + 60+\angle POQ =180\degree \\ \ \\ \angle POQ =180\degree - 120\degree = 60\degree$$​​