Ukraine 8 клас Geometry
Chapters: 5
1. Quadrilaterals
Polygons and angle sum · Parallelogram · Rectangle, rhombus, square · Midline of a triangle · Trapezium · Central and inscribed angles · Inscribed and circumscribed quadrilaterals
- Quadrilaterals (4-gons) – A quadrilateral (4-gon) has 4 sides and 4 angles that add to 360°. In a parallelogram, opposite sides are parallel and equal, opposite angles are equal, and the diagonals cut each other in half. Each of these facts also works as a test. The segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long. The three medians of a triangle meet at one point that cuts each median in the ratio 2 : 1.
- Circles: Chords and Angles – A chord joins two points on a circle. Longer chords make bigger angles at the centre, and equal chords make equal angles. The perpendicular from the centre to a chord cuts it in half, and equal chords are the same distance from the centre. The angle an arc makes at the centre is double the angle it makes anywhere on the rest of the circle, so the angle in a semicircle is 90°. In a cyclic 4-gon, opposite angles add to 180°.
2. Similarity of triangles
Proportional segments · Similar triangles · Tests for similarity
- Similar Triangles – Two figures are similar when they have the same shape but maybe a different size. Two triangles are similar if their matching angles are equal and their matching sides are in the same ratio. Basic Proportionality Theorem (BPT): a line parallel to one side of a triangle cuts the other two sides in the same ratio; its converse is also true. You can prove similarity with AA, SSS or SAS.
3. Solving right triangles
Metric relations in a right triangle · Pythagorean theorem · Trigonometric ratios of acute angles
- Similar Triangles – Two figures are similar when they have the same shape but maybe a different size. Two triangles are similar if their matching angles are equal and their matching sides are in the same ratio. Basic Proportionality Theorem (BPT): a line parallel to one side of a triangle cuts the other two sides in the same ratio; its converse is also true. You can prove similarity with AA, SSS or SAS.
- Pythagoras Theorem – In any right-angled triangle, the square on the longest side (the hypotenuse) equals the sum of the squares on the other two sides: a² + b² = c².
4. Geometric quantities: areas
Degree measure of an arc · Area of polygons
- Circles: Chords and Angles – A chord joins two points on a circle. Longer chords make bigger angles at the centre, and equal chords make equal angles. The perpendicular from the centre to a chord cuts it in half, and equal chords are the same distance from the centre. The angle an arc makes at the centre is double the angle it makes anywhere on the rest of the circle, so the angle in a semicircle is 90°. In a cyclic 4-gon, opposite angles add to 180°.
5. Geometry problems for studying real processes
Practical geometry problems
- Mathematical Modelling: Using Maths to Describe the Real World – A mathematical model is an equation, graph or table that describes a real situation in a simple way. The modelling cycle: understand the real problem → choose variables and make assumptions → build a model (for example linear, quadratic or exponential) → solve and predict → check the answer against real data → improve the model or state its limits. No model is perfect; a good one is simple and close enough to be useful.