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Geometry, grade 10 (5 units)

Plane geometry at the 5-unit level, and an introduction to analytic geometry.Subjects

Every theorem is proved, and a proof rests only on definitions and on theorems proved before it. The course goes on from the geometry of grades 8 and 9 – congruent triangles, the isosceles triangle, parallel lines and the quadrilaterals – and from the linear function and the parabola. The topics, in order:

  • an introduction to analytic geometry: the distance and the midpoint, the slope and the equation of a line, parallel and perpendicular lines, the equation of a circle, and a line meeting a parabola and a circle;
  • triangles: the midsegment of a triangle and of a trapezoid, the meeting point of the medians, loci – the perpendicular bisector and the angle bisector, the circumscribed and the inscribed circle, and the fourth congruence theorem;
  • proportion and similarity: Thales’ theorem, its converse and its extensions, the angle bisector theorem, the similarity theorems, and the ratios of heights, medians, angle bisectors, perimeters and areas in similar triangles;
  • the circle: arcs, chords and angles in a circle, triangles and quadrilaterals inscribed in a circle, the tangent to a circle, triangles and quadrilaterals circumscribing a circle, and similar triangles in a circle.

There are two kinds of lessons:

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

How the page works: Using a page.

  1. The distance between points and the midpointLearningAnalytic geometry: a point as a pair of numbers; the distance between two points by Pythagoras' theorem, the midpoint of a segment and why it is the mean of the ends, and a point at equal distances from two points.
  2. The slope and the equation of a lineLearningThe slope of a line through two points; the equation of a line by its slope and a point and by two points; the explicit and the general equation of a line, what each shows and what each leaves out; a line parallel to the y-axis, which is no function's graph; a line with a parameter.
  3. Parallel lines, perpendicular lines and meeting pointsLearningTwo lines by their slopes – parallel when the slopes are equal, perpendicular when their product is −1, and why; the meeting point of two lines as the solution of a system; the foot of a perpendicular and the distance from a point to a line; a parameter that decides how two lines lie.
  4. The equation of a circleLearningThe canonical circle x²+y²=R² from the distance between two points – a curve that is no function's graph; a circle with any centre as the canonical circle shifted; reading the centre and the radius, a point on, inside or outside a circle, and finding a circle's equation.
  5. A line meeting a parabola and a circleLearningThe meeting points of a line and a parabola, and of a line and a circle, as the solutions of a system of a quadratic and a linear equation; two points, one point – a tangent – or none, by the discriminant; a line with a parameter that touches a parabola; the chord a circle cuts off a line.
  6. Practice: analytic geometryPracticeA family of lines with a parameter, a triangle's altitude and area in coordinates, a circle and the chord it cuts off the y-axis, a rhombus from its diagonal, a parabola with a line and a tangent, and the points at equal distances from two points.
  7. The midsegment of a triangle and of a trapezoidLearningThe theorems of grades 8 and 9 that the proofs rest on, and how a proof is written; a triangle's midsegment is parallel to the third side and half of it – the proof by congruence – and its converses; a trapezoid's midsegment is parallel to the bases and half their sum.
  8. The meeting point of the mediansLearningThe three medians of a triangle meet at one point, which divides each of them in the ratio 2:1 – the proof by midsegments; the meeting point of the medians in coordinates; proofs and calculations with it.
  9. LociLearningA locus: all the points with a property, and only they – proved in both directions; the circle, the perpendicular bisector of a segment, the angle bisector, and lines parallel to a line; the perpendicular bisector in coordinates.
  10. The circumscribed circle, the inscribed circle and the altitudesLearningThe perpendicular bisectors of a triangle meet at the centre of its circumscribed circle, the angle bisectors at the centre of its inscribed circle – both proved by loci; where the circumcentre lies; the altitudes meet at one point; constructions with ruler and compasses; the special points in coordinates.
  11. The fourth congruence theorem and the converse of PythagorasLearningTwo sides and the angle opposite the larger of them – the fourth congruence theorem, proved by contradiction; why the angle opposite the smaller side is not enough, and what it does give; the converse of Pythagoras' theorem and its proof.
  12. Practice: triangles, midsegments and lociPracticeThe four triangles the midsegments make, the meeting point of the medians, a midsegment in a right-angled triangle, a point at equal distances from two points, the circumscribed and the inscribed circle of an isosceles triangle in coordinates, and equal altitudes by the fourth congruence theorem.
  13. Proportion and Thales' theoremLearningProportional segments and the properties of a proportion; parallel lines that cut equal segments on one arm of an angle cut equal segments on the other; Thales' theorem; dividing a segment into equal parts and in a given ratio.
  14. The extended Thales theorem and its converseLearningA line parallel to a side of a triangle cuts the other two sides or their extensions proportionally; the ratio of the parallel segment to the side, DE:BC = AD:AB; the converse of Thales' theorem.
  15. The angle bisector theoremLearningAn internal angle bisector of a triangle divides the opposite side in the ratio of the sides enclosing the angle – the proof by Thales' theorem – and the converse; calculations and proofs.
  16. The similarity theoremsLearningSimilar triangles and the similarity ratio; the AA similarity theorem – the proof by Thales' theorem – and the SAS and SSS similarity theorems; proofs with similarity, and the altitude to the hypotenuse of a right-angled triangle.
  17. Ratios in similar trianglesLearningIn similar triangles the ratio of corresponding heights, medians, angle bisectors and perimeters equals the similarity ratio, and the ratio of the areas is its square; similar polygons.
  18. Practice: proportion and similarityPracticeThales' theorem and its converse, the angle bisector theorem, proofs with similarity, the ratio of the areas, and a trapezoid whose diagonals cross.
  19. Arcs, chords and central anglesLearningThe circle as a locus and its parts – radius, chord, diameter, arc, central angle; equal central angles, equal arcs and equal chords; the perpendicular from the centre to a chord and its converse; a chord's distance from the centre.
  20. The inscribed angleLearningAn inscribed angle is half the central angle on the same arc; inscribed angles on the same arc are equal; equal inscribed angles, equal arcs and equal chords; the inscribed angle on a diameter is right, and its converse.
  21. Triangles and quadrilaterals inscribed in a circleLearningOne circle and only one through three points not on a line – every triangle is inscribed in a circle; in a quadrilateral inscribed in a circle the opposite angles add up to 180°, and the converse; the exterior angle of such a quadrilateral; a trapezoid inscribed in a circle is isosceles.
  22. The tangent to a circleLearningA tangent has one point in common with the circle; it is perpendicular to the radius at the point of tangency, and the converse; two tangents from one point are equal, and the segment to the centre bisects the angle between them; the angle between a tangent and a chord; constructing tangents.
  23. Triangles and quadrilaterals circumscribing a circleLearningEvery triangle circumscribes a circle, centred where the angle bisectors meet; the tangent segments from the vertices; the radius of the circle inscribed in a right-angled triangle; in a quadrilateral circumscribing a circle the sums of opposite sides are equal, and the converse.
  24. Similar triangles in a circleLearningSimilar triangles found by the angles of a circle; the products of the parts of two chords that meet, of two secants from one point, and of a tangent and a secant; the radii of the circumscribed and of the inscribed circles of similar triangles are in the similarity ratio.
  25. Practice: the circlePracticeA tangent parallel to a chord, a chord and its distance from the centre, two tangents at 60°, an inscribed quadrilateral with a bisected angle, chords and secants, and the altitude to the diameter.
  26. Review: the whole coursePracticeAll of it together: a median and the meeting point of the medians in coordinates, a circle and a line, a midsegment and medians, the angle bisector theorem with Thales, similar triangles' areas and perimeters, loci and the fourth congruence theorem, and a tangent with similar triangles.
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