Question

In the diagram, \(\mathrm{O}\) is the centre of the circle, \(\mathrm{AC}\) is the diameter and if \(\angle \mathrm{APB}=120^{\circ}\), then find \(\angle \mathrm{BQC}\) (1) \(30^{\circ}\) (2) \(150^{\circ}\) (3) \(90^{\circ}\) (4) \(120^{\circ}\)

Short answer

Answer: The measure of angle BQC is \(30^{\circ}\).

Step-by-step solution

Label the given information

Let's label the center of the circle as O, the diameter as AC, the angle APB as 120°, and the unknown angle as BQC.

Identify angle AOC

Since the angle APB is 120° and angle AOB is the angle subtended by the diameter AC at point P, we know that \(\angle AOC = 180° - \angle APB = 180° - 120° = 60°.\)

Use triangle properties to identify angle BOC

In right-angled triangle AOC, the angles inside the triangle must add up to 180°. So, we have \(\angle BOC = 180° - \angle AOC - \angle ACO = 180° - 60° - 90° = 30°.\)

Use the property that angles at the circumference on the same arc are equal

Since angle BOC is formed by the arc BC on the circumference, angle BQC is also formed by the same arc BC. Thus, according to the “angles at the circumference on the same arc” property, \(\angle BQC = \angle BOC = 30°\) So the correct answer is: (1) \(30^{\circ}\)

Key concepts

Circle Theorems

Understanding circle theorems is essential for solving complex geometry problems involving circles. These theorems are specific rules that describe the relationships between angles, radii, and chords in a circle. For instance, a foundational theorem states that the angle at the center of a circle is twice the angle at the circumference when both angles stand on the same arc. That is why in the presented problem when the angle APB is given as 120°, the angle AOC formed at the center is double that of the angle APC formed at the circumference.

Another important theorem states that the sum of the angles in a triangle must equal 180°. This fact is used when analyzing triangle AOC, which is right-angled at point C since AC is a diameter. Applying these rules reduces complexity and leads to the correct calculation of angle BQC, as seen in the solution steps provided.

Inscribed Angles

Inscribed angles, also known as angles at the circumference, are angles that have their vertex on the circle and are formed by two intersecting chords. The inscribed angle theorem is particularly useful and asserts that an inscribed angle is always half the measure of the central angle subtended by the same arc. For instance, the angle APB in the exercise is an inscribed angle subtended by the arc AB.

This theorem explains why when working with inscribed angles in circle geometry problems, we can make direct connections to angles at the center and understand their relationships. Furthermore, implicitly used in the problem is the fact that all angles inscribed on the same arc are equal. Consequently, angle BQC, which is another angle inscribed on the same arc BC as BOC, will be equal to angle BOC.

Circumference Angles

Circumference angles, or angles at the circumference, are influenced by another key circle theorem known as the Alternate Segment Theorem. This theorem explains that the angle between the tangent and the chord through the point of contact is equal to the angle in the alternate segment. While the initial problem didn't include tangents, understanding this theorem is part of comprehending the broader context of angles in circle geometry.

Further exploring our given problem, we know that the solution leveraged the property of circumference angles—specifically, that angles subtended by the same arc at the circumference are equal. In simpler terms, if two angles sit on the same arc slice of the circle's edge, like angles BQC and BOC, they must be equal in measure. Recognizing these relationships helps us to eventually determine that the answer to the problem is that angle BQC is indeed 30°, consistent with the information given and the rules of circle geometry.