Circles and Angle Properties
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Circles and Angle Properties — Free MYP5 Mathematics (Standard) Practice Questions

1QuestionAngles in the Same Segment and Alternate Segment TheoremConcept Practice
2 marks~3 minCriterion A
A satellite dish engineer checks that a circular signal reflector is correctly aligned. The circle has centre OO. Points AA, BB, and CC lie on the circumference. The engineer measures ∠ABC=32°\angle ABC = 32°.
a
Calculate ∠AOC\angle AOC. [1]
b
The alignment specification requires the central angle ∠AOC\angle AOC to be greater than 60°60°. Justify whether the reflector meets this specification, using your result from part (a). [1]
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2QuestionAngles in the Same Segment and Alternate Segment TheoremAssessment Practice
2 marks~3 minCriterion D
A surveyor places two observation points, A and B, on what is assumed to be a circular arc around the base of a distant communication tower. Using a theodolite, the surveyor measures the angle to the tower from each point, applying the angles in the same segment theorem to determine the tower's position.

Justify whether uneven terrain at the survey site undermines the reliability of the tower's calculated position. [2]
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Solutions

3QuestionAngles at the Centre and in the CircleAssessment Practice
4 marks~6 minCriterion B
A circular window in an architect's design is divided into four sections by a cyclic quadrilateral ABCDABCD, where each vertex lies on the circle. The architect claims the design has structural symmetry only if opposite angles are supplementary.

Given that ∠DAB=73°\angle DAB = 73° and ∠BCD=x°\angle BCD = x°, study the diagram.
a
State the theorem that relates opposite angles in a cyclic quadrilateral. [1]
b
Calculate the value of xx. [1]
c
Justify, using the relationship between inscribed angles and their subtended arcs, why opposite angles in any cyclic quadrilateral must always be supplementary, and hence assess whether the architect's symmetry condition is always satisfied. [2]
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Solutions

4QuestionAngles at the Centre and in the CircleAssessment Practice
5 marks~8 minCriterion C
A clock face has centre OO and radius rr. At 2:00 PM the tip of the hour hand is at point AA; at 5:00 PM it has moved to point BB, both points lying on the circumference. Point CC also lies on the circumference of the clock face.
a
Calculate the angle θ=∠AOB\theta = \angle AOB, in degrees, swept by the hour hand between 2:00 PM and 5:00 PM. [1]
b
Given that CC lies on the minor arc ABAB, deduce the angle ∠ACB\angle ACB. [2]
c
A clock designer claims that any position of CC on the major arc ABAB gives an obtuse ∠ACB\angle ACB. Justify whether this claim is correct. [2]
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Solutions

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