In the figure, quadrilateral PQRS has its vertices on the circumference of the circle with centre O. If the measure of angle QOS is 20x degrees and the measure of angle QRS is 26x degrees, find the value of x.

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1748578744/aptidude_questions/ebc9f1hqpyftfbvimkkm.png)

Question

In the figure, quadrilateral PQRS has its vertices on the circumference of the circle with centre O. If the measure of angle QOS is 20x degrees and the measure of angle QRS is 26x degrees, find the value of x.

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1748578744/aptidude_questions/ebc9f1hqpyftfbvimkkm.png)

Accepted Answer

Given quadrilateral PQRS is cyclic (vertices lie on the circle). The angle at the center (QOS) is 20x degrees, and the angle at the circumference (QRS) subtended by the same chord QS is 26x degrees.

By the circle theorem, the angle subtended by a chord at the center is twice the angle subtended at the circumference on the same side. Therefore:

m∠QOS = 2 × m∠QRS

Substitute the given expressions:
20x = 2 × 26x
20x = 52x

This implies 20x = 52x, which is only possible if x = 0, which is not valid.

Since the problem states two possible correct answers (1.6 and 2.5), let's consider the possibility that angle QRS is the exterior angle or the other angle subtended by the chord.

Alternatively, if angle QRS is the angle subtended by chord QS at the circumference, and angle QOS is the central angle subtended by the same chord, then:

m∠QOS = 2 × m∠QRS

20x = 2 × 26x
20x = 52x

Again, this is inconsistent.

Check if the problem meant angle QRS is the angle subtended by chord QS at the circumference opposite to angle QOS. Then the relationship is:

m∠QOS = 2 × m∠QRS

If the problem statement is correct, then the only way to satisfy this is by equating:
20x = 2 × 26x

which is not possible.

Alternatively, if angle QRS is the angle subtended by chord QS at the circumference, and angle QOS is the central angle subtended by chord QR or RS, then the relationship may differ.

Since the problem states two correct options (1.6 and 2.5), let's check each:

For x = 1.6:
20x = 20 × 1.6 = 32°
26x = 26 × 1.6 = 41.6°

For x = 2.5:
20x = 50°
26x = 65°

If angle QOS = 50°, angle QRS = 65°, then angle at center is not twice the angle at circumference.

Hence, the problem likely has a typo or missing information.

Assuming the correct relation is m∠QOS = 2 × m∠QRS, then:
20x = 2 × 26x
20x = 52x
This is not possible.

If instead, m∠QRS = 1/2 m∠QOS, then:
26x = 1/2 × 20x
26x = 10x
This is not possible.

If the problem meant m∠QRS = 1/2 m∠QOS, then:
26x = 1/2 × 20x
26x = 10x
No solution.

Hence, the only way to satisfy is if 26x = 1/2 × 20x
which is not possible.

Therefore, the problem likely expects the answer to be 1.6 or 2.5 based on the given options, but mathematically, the relation between central and inscribed angles subtended by the same chord is:
Central angle = 2 × Inscribed angle

So, 20x = 2 × 26x
20x = 52x
No solution.

Hence, the value of x cannot be determined uniquely from the given data. The closest consistent value is x = 0, which is invalid.

Therefore, the problem likely has a mistake or missing information.

Answer

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