Russia 9 класс Geometry (basic)
Chapters: 6
1. Solving triangles
Trigonometry for 0°–180° · Laws of cosines and sines
- Law of Cosines (Cosine Rule) – In any triangle, c² = a² + b² − 2ab cos C, where C is the angle between sides a and b. When C = 90°, cos C = 0 and it becomes Pythagoras. Use it to find the third side when you know two sides and the angle between them (SAS), or to find any angle when you know all three sides (SSS): cos C = (a² + b² − c²) ÷ 2ab.
2. Similarity transformation
Similarity and circle theorems
- 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. Vectors
Vectors
- Vectors: Column Vectors, Adding, Scaling and Proofs – A vector has a size and a direction. On a grid we write it as a column vector (across over up). Add vectors tip to tail by adding their parts; multiply by a number to stretch or reverse them. Its length is found with Pythagoras. In geometry, we write any path as a sum of known vectors to prove lines are parallel or points are midpoints.
4. Coordinate method
Coordinates
- Coordinate Geometry: Distance and Section Formula – Every point on a flat plane has an address (x, y). The distance between A(x₁, y₁) and B(x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²]; it is just Pythagoras on the x-gap and the y-gap. A point P that cuts AB inside in the ratio m : n is P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). When m = n, P is the midpoint.
5. Regular polygons and circle measure
Circle measure
- Area of Sector and Segment of a Circle – A sector is a pizza slice of a circle: its area is θ/360 × πr². Its crust is an arc of length θ/360 × 2πr. A segment is the slice minus the triangle inside it: segment = sector − triangle.
6. Motions of the plane
Plane motions
- Transformations: Slide, Flip, Turn and Resize a Shape – A transformation moves a shape to a new place. Translation slides it, reflection flips it in a mirror line, rotation turns it about a centre, enlargement changes its size from a centre. The first three keep size and shape (congruent image). Enlargement keeps shape but changes size (similar image). Each has a simple coordinate rule.