Circle Geometry Riders: Proofs, Converses & Method (Grade 11)

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Grade 11 · Euclidean Geometry

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/ 30

Answer all questions. Show all your working — marks are awarded for method as well as the final answer.

1
State the axiom about a tangent and a radius that all of circle geometry is built on.
(2)
2
O is the centre and AP̂B = 28°, with A, B and P on the circle. Determine AÔB, with a reason.
(2)
3
In cyclic quadrilateral ABCD, Â = 118°. Determine Ĉ, and state the theorem you used.
(3)
4
In quadrilateral KLMN, K̂ = 96° and M̂ = 84°. Prove that KLMN is cyclic.
(3)
5
O is the centre of a circle of radius 17 cm. A chord PQ is 30 cm long and M is its midpoint. Determine OM.
(4)
6
TA is a tangent at A and AB is a chord. The tangent–chord angle TÂB = 52°. Write down the size of the angle in the alternate segment, with a reason, and then determine OÂB.
(3)
7
Explain why you may NOT quote “opp ∠s of a cyclic quad” as a reason inside a proof that a quadrilateral IS cyclic.
(3)
8
In △OMA, O is the centre, OA is a radius and OM ⊥ AB where AB is a chord. Prove that AM = MB.
(4)
9
State the converse of the tangent–chord theorem and say what it is used to prove.
(3)
10
O is the centre and AÔB = 96°. OA and OB are radii. Determine OÂB, with reasons.
(3)

TOTAL: 30 marks