In the diagram POQ is a diameter of the circle PQRS….

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:

1. Option A is incorrect.
2. Option B is incorrect.
3. C is incorrect.
4. 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 /