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Geometry, grade 10 (4 units)

Three fields of geometry, learnt together: plane geometry, trigonometry and analytic geometry.Subjects

The course goes on from the geometry of grades 8 and 9 – congruent and similar triangles, Pythagoras’ theorem, the quadrilaterals and the median to the hypotenuse – and from the linear function of grade 8. The topics, in order:

  • analytic geometry: the distance between points, the midpoint of a segment, the slope and the equation of a line, parallel and perpendicular lines, and where two lines meet;
  • similar triangles: the similarity theorems, the midsegment of a triangle and of a trapezoid, the ratio of perimeters, of areas and of special lines;
  • trigonometry: sine, cosine and tangent, the exact values for 30∘\displaystyle 30^{\circ}, 45∘\displaystyle 45^{\circ} and 60∘\displaystyle 60^{\circ}, the sine of an obtuse angle, a triangle’s area, shapes that split into right-angled triangles, and the slope as the tangent of an angle;
  • special lines in a triangle: the medians and where they meet, the angle bisectors and the angle bisector theorem, the perpendicular bisectors and the altitudes.

A result you go on working with is written exactly: sin⁡45∘=22\displaystyle {\sin45^{\circ}}={\frac{\sqrt{2}}{2}}, not 0.707\displaystyle 0.707; 10\displaystyle \sqrt{10}, not 3.16\displaystyle 3.16.

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 midpoint of a segmentLearningA segment's length from the coordinates of its ends – a segment parallel to an axis, and any segment, by Pythagoras' theorem; the midpoint of a segment, and one end of a segment from the other end and the midpoint.
  2. The slope and the equation of a lineLearningThe slope of a line through two points, a line's equation in explicit and in implicit form, lines parallel to the axes, and finding a line's equation from its slope and a point, and from two points.
  3. Parallel lines, perpendicular lines and meeting pointsLearningWhere a line crosses the axes, how two lines lie by their slopes – the same line, parallel, crossing and perpendicular – and the meeting point of two lines by a system of equations.
  4. Practice: points, segments and linesPracticeA parallelogram and a rectangle in coordinates, the midsegment of a triangle whose vertices are given, the equations of parallel and perpendicular lines, and a point at equal distances from two points.
  5. Similar triangles: the similarity theorems and proofsLearningSimilar triangles and the three similarity theorems again – AA, SAS and SSS – writing a similarity in the right order, proving triangles similar and finding sides, and the altitude to the hypotenuse of a right-angled triangle.
  6. The midsegment of a triangle and of a trapezoidLearningA triangle's midsegment is parallel to the third side and half of it – the proof by similarity – and the converse theorems; a trapezoid's midsegment is parallel to the bases and half their sum; the quadrilateral whose vertices are the midpoints of a quadrilateral's sides.
  7. The ratio of perimeters, areas and special linesLearningIn similar triangles the ratio of the perimeters is the similarity ratio, the ratio of the areas its square, and the ratio of corresponding altitudes, medians and angle bisectors is the similarity ratio as well: a proof, calculations and a rectangle inscribed in a triangle.
  8. Practice: similar triangles and midsegmentsPracticeA trapezoid inside an isosceles triangle, a parallelogram in coordinates, the diagonals of a trapezoid, the ratio of areas a midsegment cuts off, and the altitude to the hypotenuse – similarity proofs and calculations.
  9. The sine, cosine and tangent of an acute angleLearningThe trigonometric functions as ratios of sides in a right-angled triangle, why they depend only on the angle, their values as the angle grows from 0° to 90°, the angles 0° and 90°, and finding an angle from its sine, cosine or tangent.
  10. Exact values and links between the functionsLearningThe sine, cosine and tangent of 30°, 45° and 60° exactly – from an isosceles right-angled triangle and half an equilateral triangle; the tangent as the sine over the cosine, sin²α+cos²α=1, and sin(90°−α)=cos α.
  11. Calculations in a right-angled triangleLearningFinding a side, an angle, the perimeter and the area of a right-angled triangle with sine, cosine and tangent, Pythagoras' theorem and the triangle's properties – an exact result for the special angles, and rounding only at the end.
  12. The sine of an obtuse angle and a triangle's areaLearningThe sine of an obtuse angle is defined so that sin(180°−α)=sin α; two angles between 0° and 180° with the same sine; and a triangle's area from two sides and the angle between them, S=½ab·sinγ.
  13. Shapes that split into right-angled trianglesLearningThe trapezoid, the rhombus, the rectangle and the parallelogram: altitudes and diagonals that split them into right-angled triangles, and finding lengths, angles, perimeters and areas by trigonometry and the properties of the quadrilaterals.
  14. The slope as the tangent of an angleLearningA line's slope is the tangent of the acute angle between it and the x-axis – with a plus sign for a rising line and a minus sign for a falling one; the angle of a line, a line from its angle, and the angles of a triangle in coordinates.
  15. Practice: trigonometryPracticeA rectangle and a triangle in coordinates – equations, angles, the perimeter and the area in more than one way; an isosceles triangle, an area with an obtuse angle, exact values and a rhombus from its diagonals.
  16. The medians and where they meetLearningThe three medians of a triangle meet at one point, which divides each median in the ratio 2:1 – the proof by a midsegment and similarity; a median splits the triangle into two triangles of equal area; where the medians meet, in coordinates.
  17. Angle bisectors and the angle bisector theoremLearningThe angle bisector as the locus of the points at equal distances from the arms of the angle, where the angle bisectors of a triangle meet, and the theorem of an interior angle bisector in a triangle – BD:DC=AB:AC – and its converse.
  18. Perpendicular bisectors and altitudesLearningThe perpendicular bisector as the locus of the points at equal distances from the ends of a segment, where a triangle's perpendicular bisectors meet, where the altitudes meet in an acute, a right-angled and an obtuse triangle, and the equations of perpendicular bisectors and altitudes in coordinates.
  19. Practice: special lines in a trianglePracticeAn angle bisector in coordinates, where the medians meet and the ratio 1:3, medians and altitudes in an isosceles triangle, the angle bisector theorem, and a point at equal distances from two points – proofs and calculations.
  20. Review: the whole coursePracticeAll of it together: a parallelogram in coordinates, similar triangles, trigonometry in an isosceles triangle and in a parallelogram, where the medians meet, the angle bisector theorem, where the perpendicular bisectors of a right-angled triangle meet, and a line from its angle.
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