Title price = 99 $
 Chapter price = 10 $
ALGEBRA 1 Version 2009
Revised and highly enlarged.
Symmetries, Thales, Pythagoras, Figures, constructions
Middle school 11-15 years


  Click arrows to display the contents of chapters
 Click an icon signaling courses and exercises which can be downloaded
  Click on a box or boxes corresponding to the chapter (s) you want to buy

   1. Using tools to draw lines and circles.
  •           1. Drawing a line, a segment, a half-line, a triangle.
  •           2. Drawing an ordinary triangle.
  •           3. Definition of a circle, of a disk. Drawing a circle and an arc of a circle.
  •           4. Drawing a triangle given the lengths of its sides.
  •           5. Drawing isosceles and equilateral triangles.
  •           6. Drawing triangles with given lengths of their sides: ordinary, isosceles, equilateral.
  •           7. Free drawing.
   2. Orthogonal median of a segment. Orthogonal medians of a triangle and round circle.
  •           1. Construction and properties of an orthogonal median to a segment.
  •           2. Review.
  •           3. Property of the line joining the center of two intersecting circles.
  •           4. Orthogonal medians of a triangle are convergent. Circle round a triangle.
  •           5. Drawing a circle round an ordinary triangle.
  •           6. Circle round a right-angled triangle.
  •           7. Drawing the perpendicular to a line using a ruler and compasses only.
  •           8. Free drawing.
   3. The angles: definition, equality, bisector, measure using the protractor.
  •           1. Definition of an angle. Use of the protractor. Particular angles.
  •           2. To name the displayed angles.
  •           3. Measuring the triangle angles.
  •           4. To draw a triangle knowing one side cornered between two given angles.
  •           5. Equality and sum of two angles. Angle bisector: definition and drawing.
  •           6. Drawing the bisectors of an angle and then of a triangle.
  •           7. Supplementary angles. Properties of vertically opposite angles.
  •           8. Drawing bisectors of two adjacent supplementary angles. Properties of these bisectors.
  •           9. Free drawing.
   4. Symmetry with respect to an axis or axial symmetry.
  •           1. Definition of the symmetrical point with respect to an axis.
  •           2. Geometrical construction of the symmetrical of a point with respect to an axis.
  •           3. Symmetrical of a line (L) with respect to an axis (D).
  •           4. Construction of the symmetrical of a line. Lines invariant by axial symmetry.
  •           5. Symmetrical of a circle with respect to an axis. Symmetry axes.
  •           6. Symmetrical of a triangle with respect to one of its sides and to one of its perpendicular heights.
  •           7. Axial symmetry preserves distances and absolute values of angle measures.
  •           8. Symmetry axis of isosceles triangle.
  •           9. Symmetry axes of equilateral triangle.
  •           10. To mark a point on a figure and find its symmetrical on the symmetrical figure.
  •           11. I draw a figure; SD operator reproduces its symmetrical.
   5. Symmetry with respect to a point or central symmetry.
  •           1. Definition of the symmetrical of a general point with respect to a given point.
  •           2. The symmetrical of a line with respect to a point O.
  •           3. To draw line (L?) symmetrical of line (L) with respect to a point.
  •           4. Applying symmetry to draw a line (L?), parallel to a line (L) passing through a point A.
  •           5. The symmetrical of a triangle with respect to one of its vertices.
  •           6. The symmetrical of a triangle with respect to the middle of one side.
  •           7. The symmetrical of a circle with respect to a point.
  •           8. The symmetrical of the center of the round circle of a triangle with respect to a vertex.
  •           9. To mark a point on a figure and find its symmetrical on the drawn symmetrical figure.
  •           10. Free drawings: draw any figure, and then operator SO will draw its symmetrical.
   6. Angles formed by a line crossing two parallels. The sum of a triangle?s angles is 180.
  •           1. Perpendicular to a line drawn from a point. Distance to the line.
  •           2. To construct the symmetry axis of a line passing through a given point.
  •           3. Parallel and perpendicular lines. Euclid?s postulate.
  •           4. Angles formed by a line crossing two parallels are equal or supplementary.
  •           5. Bisectors properties of two alternate or corresponding angles.
  •           6. The sum of a triangle angles is 180. Particular triangles.
  •           7. To construct a triangle given its perimeter and two of its angles.
  •           8. Definition of a polygon and a convex polygon.
  •           9. The sum of a convex polygon angles, with n sides, is (n-2)180.
  •           10. To measure a pentagon?s angles and compute their sum.
   7. Definition and properties of a parallelogram.
  •           1. A quadrilateral with a center of symmetry: the parallelogram.
  •           2. Angular properties of a parallelogram.
  •           3. Opposite sides of a parallelogram are equal in pairs.
  •           4. To draw from a point, a parallel to a line by constructing a parallelogram.
  •           5. Definition of a vector and equal vectors. How vectors are linked with the parallelogram.
  •           6. Five properties permit to recognize a parallelogram.
  •           7. Two parallelograms with the same center of symmetry.
  •           8. Free drawing.
   8. From the parallelogram to the rectangle, the square, the diamond. Trapezoids.
  •           1. The rectangle, a parallelogram with a right angle.
  •           2. The rectangle, a parallelogram with equal diagonals. Circle round a rectangle.
  •           3. The rectangle possesses two orthogonal symmetry axes.
  •           4. Rectangles drawn within two intersecting circles.
  •           5. The diamond is a parallelogram with two consecutive equal sides.
  •           6. The square possesses the properties of both the rectangle and the diamond.
  •           7. Ordinary trapezoid.
  •           8. Properties of rectangle and isosceles trapezoids.
  •           9. Circle round an isosceles trapezoid.
   9. To recognize particular quadrilaterals by their properties.
  •           1. Properties characterizing particular quadrilaterals.
  •           2. Necessary and sufficient conditions for a quadrilateral to be a parallelogram.
  •           3. Necessary and sufficient conditions for a quadrilateral to be a rectangle.
  •           4. Necessary and sufficient conditions for a quadrilateral to be a diamond.
  •           5. Necessary and sufficient conditions for a quadrilateral to be a square.
  •           6. Necessary and sufficient conditions for a quadrilateral to be a trapezoid.
   10. Projection onto a line parallel to a direction. Thales' theorem.
  •           1. Equidistant points on a line have equidistant projections.
  •           2. Applying projection to divide a segment into equal segments.
  •           3. Properties of a segment joining the middles of a triangle's sides.
  •           4. Thales' theorem.
  •           5. Allowed operations on two equal ratios. Transposition along the diagonals.
  •           6. Useful relations resulting from proportionality.
  •           7. Similar triangles. Sides' proportionality. Thales' reciprocal theorem.
  •           8. How to determine the missing lengths using Thales' theorem.
  •           9. A parallelogram inscribed into a quadrilateral.
   11. Convergent lines in a triangle: orthocenter, center of gravity...
  •           1. Perpendicular heights of a triangle converge at a point called the orthocenter.
  •           2. Triangle medians converge at a point called the center of gravity.
  •           3. Triangle internal bisectors converge at the center of the inscribed circle.
  •           4. Triangle internal and external bisectors converge at the centers of externally inscribed circles.
  •           5. Use compasses to draw the perpendicular to a line from a point.
  •           6. To draw the bisector of an angle with an inaccessible vertex.
  •           7. Sum of the distances from a point to an equilateral triangle.
   12. Relative positions of a line and a circle, and of two circles. Central and inscribed angles.
  •           1. Tangent to a curve at a point.
  •           2. The tangent to a circle is perpendicular to the radius. Relative positions of a line and a circle.
  •           3. Relative positions of two circles. Tangent, orthogonal, and concentric circles.
  •           4. Locus of the middles of a circle's chords passing through a given point.
  •           5. Central and inscribed angles in a circle.
  •           6. Interior, exterior and tangent angles to a circle.
  •           7. Drawing triangle's bisectors using the round circle.
  •           8. In soccer, shooting positions through a given angle.
   13. Rectangle and polygon's areas. Pythagoras' theorem.
  •           1. Perimeters and areas. The perimeter and area of a rectangle and of a square.
  •           2. Parallelogram area.
  •           3. Ordinary triangle area.
  •           4. Trapezoid area.
  •           5. Pythagoras' theorem and its reciprocal. A graphical representation.
  •           6. Two other proofs of Pythagoras' theorem.
  •           7. A right angled triangle, with a known hypotenuse and one side of its right angle.
  •           8. A right angled isosceles triangle and a half equilateral triangle.
  •           9. A geometrical drawing of Rac(n), the square root of an integer.
   14. Translations and 2-dimensions vectors. Distances and angles conservation.
  •           1. Vector definition. Equality of two vectors.
  •           2. Translation by a vector u: image of a point by translation.
  •           3. Translation of parallelogram vertices: to use the correct vocabulary.
  •           4. Translation of a figure by a vector u. Translation conserves distances and angles.
  •           5. Translated figures of a segment, of a half-line, of a line, and of a vector.
  •           6. Translation of a triangle and its orthogonal height.
  •           7. Free activity: to determine the translation vector given a figure and its image.
  •           8. The translated figure of a circle is a circle, with the same radius.
  •           9. The property of three circles with the same radii which intersect at the same point.
   15. Rotation round a point. Some figures which are invariant by rotation.
  •           1. Definition of a rotation round a point with angle a.
  •           2. Rotation round a point conserves distances and angles.
  •           3. Image of a line segment by rotation: angles and sides conservation.
  •           4. To find the rotation that makes two equal segments correspond.
  •           5. Image of a line by a rotation with center O and angle a.
  •           6. A figure symmetrical or invariant by rotation: the equilateral triangle.
  •           7. Symmetries of the square.
  •           8. Symmetries of the regular hexagon.
  •           9. Symmetries of regular polygons with n vertices.
  •           10. Drawing a regular octagon and using its symmetries.
   16. The circle's perimeter and definition of the number p. The disc area. The arc and angular sector.
  •           1. To discover the proportionality of the circle's perimeter to its radius.
  •           2. The perimeter of regular polygon inscribed in a circle.
  •           3. The ratio of the circle's perimeter to its diameter defines the number p.
  •           4. Runners' gaps on athletic tracks.
  •           5. The area of a disc with radius R is p times the area of a square with side R.
  •           6. Pythagoras's theorem, half-discs and Hippocrates lunes.
  •           7. Length of a circle's arc.
  •           8. The distance between two towns with the same longitude.
  •           9. The area of a circular angular sector.
  •           10. The area swept by car's windshield wiper blades.
   17. An angle's measures, definition of trigonometric functions and their relations. Applications.
  •           1. An angle's measures in radians and degrees.
  •           2. Trigonometric ratios: cosine, sine, tangent, cotangent.
  •           3. To determine the sides of a right-angled triangle given the hypotenuse and an acute angle.
  •           4. To determine the sides of a triangle given one side and two angles.
  •           5. Trigonometric ratios of particular angles.
  •           6. The slope of a line in a reference frame. Application to a road's inclines.
  •           7. The angle between a road of a given incline, and the horizontal.
  •           8. Relations between trigonometric ratios, which derive from Pythagoras theorem.
  •           9. An equilateral triangle and a square having a common side.
   18. Vectors in a plane: addition, multiplication by a number, collinearity, components.
  •           1. Definition and equality of two vectors.
  •           2. Sum of two vectors. Opposite vectors. Null vector.
  •           3. Exercises using the addition of vectors and their opposites.
  •           4. Multiplication of a vector by a number. Collinear vectors. Thales' Vector Theorem.
  •           5. Exercises which use collinearity.
  •           6. Vectors formed by linear combinations of vectors.
  •           7. Components of a vector in a basis. Properties of the components of collinear vectors.
  •           8. Components of vectors in an orthonormal basis.
  •           9. Properties of the median of any triangle. Vector properties of a parallelogram.
  •           10. Center of gravity of a triangle: Vector properties.
  •           11. Cartesian coordinates of a vector. Components of a linear combination.
   19. Isometric triangles. Similar triangles. Homothety.
  •           1. Isometric triangles. Superposable triangles. Isometric non-superposable triangles.
  •           2. The two other cases of isometry (equality) of ordinary triangles. Case of right-angled triangles.
  •           3. Exercises with isometric triangles.
  •           4. Homothety enlarges or reduces a figure while keeping the angles and gives a similar figure.
  •           5. Homothetic triangles and circles.
  •           6. Comparison of two homotheties with centers O and O1 and the same ratio k.
  •           7. Exercises with direct application of homothety.
  •           8. Determine the centers and the homothetic ratios of any two circles.
  •           9. Complements on similar triangles. The 3 cases of similarity.
  •           10. Relation between the height of a triangle, the radius of the circumscribed circle and the sides.
  •           11. Variable triangle similar to a fixed triangle, in two secant circles.