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

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Grade 11 · Euclidean Geometry

MEMORANDUM

Name:

Class:

Date:

Mark

/ 30

Answer all questions. Show all your working — marks are awarded for method as well as the final answer. Teacher copy: accept any correct equivalent method.

1
State the axiom about a tangent and a radius that all of circle geometry is built on.
(2)
A tangent to a circle is perpendicular to the radius drawn to the point of contact ✓✓(2)
2
O is the centre and AP̂B = 28°, with A, B and P on the circle. Determine AÔB, with a reason.
(2)
AÔB = 2 × 28° = 56° ✓ (∠ at centre = 2 × ∠ at circumference) ✓(2)
3
In cyclic quadrilateral ABCD, Â = 118°. Determine Ĉ, and state the theorem you used.
(3)
Ĉ = 180° − 118° ✓ = 62° ✓ (opp ∠s of a cyclic quad are supplementary) ✓(3)
4
In quadrilateral KLMN, K̂ = 96° and M̂ = 84°. Prove that KLMN is cyclic.
(3)
K̂ + M̂ = 96° + 84° ✓ = 180° ✓; ∴ KLMN is cyclic (converse: opp ∠s supplementary) ✓(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)
OM ⊥ PQ ✓ (line from centre to midpt of chord); PM = 15 cm ✓; OM² = 17² − 15² = 289 − 225 = 64 ✓; OM = 8 cm ✓(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)
The angle in the alternate segment = 52° ✓ (tan–chord ∠) ✓; OÂB = 90° − 52° = 38° ✓ (tan ⊥ radius)(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)
That reason assumes the quadrilateral is already cyclic ✓, which is exactly what is being proved ✓. You must show the opposite angles add to 180° and then conclude using the CONVERSE ✓.(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)
Join OB ✓ (construction); OA = OB (radii) ✓; OM = OM (common); OM̂A = OM̂B = 90° (given) ✓; ∴ △OMA ≡ △OMB (RHS); ∴ AM = MB (≡△s) ✓(4)
9
State the converse of the tangent–chord theorem and say what it is used to prove.
(3)
If the angle between a line and a chord equals the angle in the alternate segment ✓, then that line is a tangent to the circle ✓. It is used to PROVE that a given line is a tangent ✓.(3)
10
O is the centre and AÔB = 96°. OA and OB are radii. Determine OÂB, with reasons.
(3)
OA = OB (radii) ✓, so △OAB is isosceles and OÂB = OB̂A (∠s opp equal sides) ✓; OÂB = (180° − 96°) ÷ 2 = 42° ✓(3)

TOTAL: 30 marks