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Geometry, grade 11 (4 units)

The circle in all three fields of geometry: proofs, trigonometry and the coordinate system.Subjects

The topics, in order:

  • the circle and its equation: the basic terms, chords, a line and a circle;
  • angles in a circle: central angles, arcs and chords, the inscribed angle, an inscribed angle on a diameter;
  • polygons and circles: a triangle inscribed in a circle, a quadrilateral inscribed in a circle, the law of sines, area by the sine;
  • tangents: the tangent to a circle, the tangent’s equation, a circle tangent to the axes, a circle inscribed in a triangle;
  • the circumference, the area of a disc and the number π.

The course goes on from grade 10’s geometry (4 units): special lines in a triangle, similarity, trigonometry, lines in the coordinate system.

There are two kinds of lessons:

  • learning lessons – an explanation, worked examples and a few questions to try;
  • practice lessons – questions only.

How the page works: Using a page.

  1. The circle: definitions and termsLearningThe circle as a locus, the radius, the diameter and the chord, an arc and a central angle, the minor and the major arc, the disc and the sector, and a point inside, on or outside a circle.
  2. The equation of a circleLearningThe canonical and the general equation of a circle, the centre and the radius from the equation, a point on the circle, and a circle's equation from its centre and a point or from the ends of a diameter.
  3. Chords and the centre of the circleLearningA perpendicular from the centre bisects a chord, a line from the centre through a chord's midpoint is perpendicular to it, a chord's perpendicular bisector passes through the centre; a chord's distance from the centre, and a check by slopes.
  4. A line and a circleLearningA line meets a circle at two points, touches it or misses it: by comparing the radius with the centre's distance from the line, and by the system of the circle's and the line's equations.
  5. Practice: the circle, its equation, a line and a circlePracticeArcs and central angles, a circle's equation, chords and their distance from the centre, a point and a circle, and a line and a circle by a system of equations.
  6. Central angles, arcs and chordsLearningThe measure of an arc, the minor and the major arc; equal central angles and equal chords; the perpendicular to a chord through the centre bisects the central angle and the arc.
  7. The inscribed angleLearningAn inscribed angle is half the central angle and half the arc; inscribed angles on the same arc are equal; equal arcs, chords and inscribed angles.
  8. An inscribed angle on a diameterLearningAn inscribed angle on a diameter is a right angle, and the converse; the hypotenuse of a right-angled triangle is a diameter; a check in the coordinate system by slopes.
  9. Practice: angles in a circlePracticeCentral angles and arcs, inscribed angles, an inscribed angle on a diameter: calculations and proofs.
  10. A triangle inscribed in a circleLearningEvery triangle can be inscribed in a circle: the perpendicular bisectors meet at the centre of the circumscribed circle; where the centre is; finding the circumscribed circle in the coordinate system.
  11. A quadrilateral inscribed in a circleLearningA quadrilateral can be inscribed in a circle if and only if a pair of opposite angles adds up to 180°; a trapezoid inscribed in a circle is isosceles; an inscribed trapezoid in the coordinate system.
  12. The law of sinesLearningA side over the sine of the angle opposite it equals the diameter of the circumscribed circle: finding a side, an angle and the radius, and a triangle inscribed in a circle.
  13. Area by the sine: the triangle and the parallelogramLearningThe area of a triangle – half the product of two sides and the sine of the angle between them; the area of a parallelogram; areas of shapes by adding and subtracting triangles.
  14. Practice: the law of sines and areasPracticeThe law of sines, the radius of the circumscribed circle, the area of a triangle and a parallelogram by the sine, and a regular polygon inscribed in a circle.
  15. The tangent to a circleLearningThe definition of a tangent, the tangent perpendicular to the radius at the point of contact, two tangents from one point, parallel tangents, and calculations by Pythagoras' theorem.
  16. The tangent's equation and a circle tangent to the axesLearningThe equation of the tangent at a point of a circle by the perpendicularity condition, horizontal and vertical tangents, a tangent as the one solution of a system, and a circle tangent to one axis or to both.
  17. A circle inscribed in a triangleLearningThe three angle bisectors meet at the centre of the inscribed circle, the tangent segments from the vertices, the inscribed circle's radius in a right-angled triangle, and a right-angled triangle in the coordinate system.
  18. Practice: tangents and the inscribed circlePracticeTangent segments, angles between tangents, a tangent's equation, a circle touching an axis, the tangent segments of a triangle and the radius of the inscribed circle.
  19. The circumference, the area of a disc and the number πLearningThe circumference and the area of a disc, the length of an arc and the area of a sector, and estimating π with regular polygons inscribed in a circle.
  20. ReviewPracticeQuestions from the whole course: chords, angles in a circle, an inscribed quadrilateral, tangents, the law of sines, area by the sine, the equation of a circle, a line and a circle, the tangent's equation and the number π.
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