Poland Liceum ogólnokształcące, klasa III Mathematics
Chapters: 3
1. Plane geometry
Circles and triangle types · Polygons and quadrilaterals · Similarity and Thales · Special points, proofs, trigonometry in figures
- Circle Theorems – Circle theorems are a small set of angle rules that are always true inside a circle. The angle at the centre is twice the angle at the edge. The angle in a semicircle is 90°. Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral add to 180°. A tangent meets the radius at 90°, two tangents from one point are equal, and the tangent–chord angle equals the angle in the alternate segment.
- Properties of Triangles and Their Centres – The angles of every triangle add up to 180°. A triangle has four famous centres. Medians meet at the centroid G, which cuts each median 2 : 1. Perpendicular bisectors meet at the circumcentre O, the centre of the circle through the corners. Angle bisectors meet at the incentre I, the centre of the circle inside that touches all sides. Altitudes meet at the orthocentre H. O, G and H lie on one line, the Euler line.
2. Analytic geometry in the plane
Lines and circles
- Straight Lines (Class 11): Slope, Forms of the Equation and Distance – The slope of a line is rise ÷ run = (y₂ − y₁)/(x₂ − x₁) = tan θ. Parallel lines have equal slopes; perpendicular lines have m₁m₂ = −1. The angle between two lines is given by tan θ = |(m₂ − m₁)/(1 + m₁m₂)|. A line can be written as y = b or x = a (parallel to an axis), y − y₁ = m(x − x₁) (point-slope), y = mx + c (slope-intercept), the two-point form, or x/a + y/b = 1 (intercept form). The distance of (x₁, y₁) from Ax + By + C = 0 is |Ax₁ + By₁ + C|/√(A² + B²).
3. Solid geometry
Lines, planes and angles in space · Volume and surface area
- Solid Geometry: Points, Lines and Planes in Space – Solid geometry studies figures in three dimensions. Three points not on one line fix a plane. Two lines in space can be parallel, intersecting or skew (not in one plane). A line can lie in a plane, cut it, or be parallel to it; it is perpendicular to a plane if it is perpendicular to two intersecting lines of that plane. Angles in space are found by projecting onto a plane and using right triangles.