Circle Theorems and Tangents
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Circle Theorems and Tangents — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionAngles at the Centre and in the CircleConcept Practice
2 marks~3 minCriterion A
A telecommunications engineer is positioning a signal repeater at point CC on the boundary of a circular coverage zone. The transmitter and receiver are fixed at points AA and BB on the boundary, with the network hub at centre OO. The engineer knows that ∠ACB=35°\angle ACB = 35°.
a
Deduce the measure of ∠AOB\angle AOB. [1]
b
The hub OO can only relay signals when the central angle ∠AOB\angle AOB is strictly less than 75°75°. Justify whether this network configuration meets the relay condition. [1]
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Solutions

2QuestionAngles in the Same Segment and Alternate Segment TheoremAssessment Practice
5 marks~8 minCriterion C
Points AA, BB, CC, DD, and EE lie on the circumference of a circle with centre OO. The line segment AEAE is a diameter. Angle BAC=34°BAC = 34°. A surveying team uses this circular arrangement to model sight-lines between observation posts, where a 90°90° angle between lines confirms a perpendicular alignment required for accurate triangulation.
a
Deduce the measure of angle BCEBCE. [1]
b
Calculate angle BECBEC and justify which circle theorem you used. [2]
c
Angle EDCEDC is claimed to equal angle BACBAC. Justify whether this claim is correct, and explain what this means for the sight-lines from posts DD and AA to the arc ECEC. [2]
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3QuestionAngles in the Same Segment and Alternate Segment TheoremAssessment Practice
5 marks~8 minCriterion B
A satellite dish is mounted on a curved support. An engineer models a cross-section of the support as a circle with centre OO. A tangent PTPT touches the circle at point AA, and point BB lies on the circumference. The angle BAP=θBAP = \theta.
a
State the size of angle OAPOAP, justifying your answer with the appropriate theorem. [1]
b
Prove that triangle OABOAB is isosceles and hence deduce angle OABOAB in terms of θ\theta. [2]
c
Prove that ∠AOB=2θ\angle AOB = 2\theta, then advise the engineer whether measuring the tangent angle θ\theta at the mounting point AA is a sufficient method for determining the central angle of the support. [2]
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4QuestionAngles in the Same Segment and Alternate Segment TheoremAssessment Practice
6 marks~9 minCriterion D
A ship's captain navigates between two coastal landmarks, AA and BB, avoiding a submerged reef. By measuring the angle θ\theta subtended by chord ABAB from the ship's position, the captain uses the angles in the same segment theorem to stay on a safe circular arc that keeps the reef outside it.
a
Explain how the angles in the same segment theorem guarantees that a constant angle θ\theta keeps the ship on the same circular arc through AA and BB. [2]
b
Explain two limitations of using angle measurements alone to maintain this safe arc in practice. [2]
c
Advise the captain whether this geometric method alone is sufficient for safe navigation, and justify your advice by recommending at least two specific additional tools or strategies that should be used. [2]
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