In the diagram POQ is a diameter of the circle PQRS. If <PSR = 145°. Find x
- A. 45°
- B. 25°
- C. 35°
- D. 55°
QUICK ANSWER…
D
SOLUTION…
If one of the sides of a triangle is the diameter of a circle, and the three angles of the triangle intersect with the circumference of the circle, then the angle opposite the diameter is always 90°, that is a rule in circle geometry. It is demonstrated below:
If PQ is the diameter, then <PSQ must be = 90° irrespective of where S touches the circumference.
If <PSQ = 90°, then <RSQ = 145 – 90 = 55°
Now there’s another rule in circle geometry, it states that if the chord of a circle makes two or more angles at any point on the circumference, these angles are absolutely the same.
For instance, in the diagram above, there are 4 chords: PQ, PS, SR, and RQ.
For chord PQ: angle PRQ and angle PSQ form a base with chord PQ, this rule states that <PRQ = <PSQ.
Similarly, for chord PS, angle PRS = angle PQS
For chord SR, angle SPR = angle SQR
And for chord RQ, angle RPQ = angle RSQ
Therefore since <RPQ = x° which is also same as <RSQ. hence, x = 55°
Now for the right answer to the above question:
- Option A is incorrect.
- Option B is incorrect.
- C is incorrect.
- D is the correct answer.
QUICK TIPS…
You may please note these/this:
- Always remember that when two angles form between the same chord and different points on the same side of the circumference of a circle, they are definitely the same.
- A chord is any straight line joining two points on the circumference of a circle.
Use the questions and answers session to deal further on this topic…
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/ culled from 2018 JAMB-UTME mathematics question 4 /
