Russia 8 класс Geometry (advanced)
Chapters: 6
1. Quadrilaterals
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.
2. Similarity
Similar triangles
- 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. Area
Areas
- Area and Perimeter: Heron's Formula, Circles and Sectors – Perimeter is the length of the edge. Area is the space inside. For a triangle with three known sides, Heron's formula gives the area without any height. For a four-sided shape whose corners sit on a circle, Brahmagupta's formula does the same. For circles, the edge is π times the diameter, and a slice (sector) is just a fraction of the whole circle.
4. Pythagorean theorem
Pythagorean theorem
- 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.
5. Trigonometry
Trigonometry of acute angle
- Trigonometric Ratios (sin, cos, tan) – In a right triangle, pick one sharp (acute) angle θ. The side facing θ is the opposite, the side touching θ (not the longest) is the adjacent, and the longest side is the hypotenuse. sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. cosec, sec and cot are their flips. The ratios depend only on the angle, not on the size of the triangle. Learn the table for 0°, 30°, 45°, 60°, 90°.
6. Circle angles and quadrilaterals
Circle theorems
- 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°.