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Circle Theorems and Tangents — Free MYP5 Mathematics (Extended) 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 . Points , , and lie on the circumference. The engineer measures .
a
Calculate . [1]
b
The alignment specification requires the central angle to be greater than . Justify whether the reflector meets this specification, using your result from part (a). [1]
Solutions
2QuestionTangent Properties and Circle Geometry ProofsAssessment Practice
2 marks~3 minCriterion C
A telecommunications engineer is positioning a satellite dish. The signal cable runs from point on the ground directly to point on the outer edge of a circular support ring, meeting the ring at exactly one point.
The diagram shows a circle with centre . The line is tangent to the circle at point .
Justify why , and explain how this perpendicularity guarantees the cable meets the ring at exactly one point. [2]
The diagram shows a circle with centre . The line is tangent to the circle at point .
Justify why , and explain how this perpendicularity guarantees the cable meets the ring at exactly one point. [2]
Solutions
3QuestionAngles 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]
Justify whether uneven terrain at the survey site undermines the reliability of the tower's calculated position. [2]
Solutions
4QuestionAngles 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 , where each vertex lies on the circle. The architect claims the design has structural symmetry only if opposite angles are supplementary.
Given that and , study the diagram.
Given that and , study the diagram.
a
State the theorem that relates opposite angles in a cyclic quadrilateral. [1]
b
Calculate the value of . [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]
Solutions
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