Ukraine 9 клас Geometry
Chapters: 7
1. Solving triangles
Trigonometric functions of angles 0°-180° · Law of cosines · Law of sines · Solving triangles in practice
- 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.
- Law of Sines (Sine Rule) – In any triangle, each side divided by the sine of the angle facing it gives the same number: a/sinA = b/sinB = c/sinC = 2R, where R is the radius of the circle through the three corners. Use it when you know two angles and a side (AAS or ASA), or two sides and an angle that is not between them (SSA). SSA can give two triangles, one or none.
2. Regular polygons
Regular polygons and their properties · Circle length and area of a disc
- Regular Polygons and Their Circles – A regular polygon has all sides equal and all angles equal. Every regular polygon has a centre O, a circumscribed circle through all its vertices (radius R) and an inscribed circle touching every side (radius r, the apothem). For n sides of length a: central angle = 360°/n, interior angle = (n − 2)·180°/n, R = a / (2 sin(180°/n)), r = a / (2 tan(180°/n)), and area = ½ · perimeter · r. Special cases: triangle R = a/√3, r = a/(2√3); square R = a/√2, r = a/2; hexagon R = a, r = a√3/2.
- 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.
3. Cartesian coordinates on the plane
Distance and midpoint · Equation of a figure: circle and line
- The Cartesian Plane: Coordinates, Distance and Midpoint – Two number lines that cross at right angles turn a flat surface into a map where every point has an address (x, y). The axes meet at the origin O(0, 0) and cut the plane into four quadrants. Distance between two points comes from Pythagoras: square the x-gap and the y-gap, add, take the square root. The midpoint is the average of the x values and the average of the y values. Three points are collinear when the two short distances add up to the long one, and they make a right angle when the squares of the two short sides add up to the square of the long side.
- Equation of a Circle – A circle is the set of points at a fixed distance r from a centre (h, k). By the distance formula its equation is (x − h)² + (y − k)² = r²; with centre at the origin, x² + y² = r². Opened up, it becomes x² + y² + Dx + Ey + F = 0, with centre (−D/2, −E/2) and r² = D²/4 + E²/4 − F. A line meets a circle in 2, 1 or 0 points when the distance d from centre to line is less than, equal to or more than r. Two circles are compared by the distance between their centres.
4. Vectors on the plane
Vectors and coordinates · Operations on vectors · Dot product
- Scalars and Vectors for Motion in a Plane (Class 11) – A scalar has only size; a vector has size and direction. Vectors are equal if their size and direction match. Multiplying by a number changes the length (a negative number flips it). Vectors add tail-to-head (triangle or parallelogram law); A − B = A + (−B). Any vector in a plane is A = Ax î + Ay ĵ with Ax = A cos θ, Ay = A sin θ. A·B = AB cos θ is a scalar; A×B has size AB sin θ and is perpendicular to both.
5. Geometric transformations
Motions (isometries) · Similarity transformation
- 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.
- 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.
6. Geometric quantities
Area formulas for a triangle
- 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.
7. 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.