Basic Geometric Concepts

 Basic Geometric Concepts

- Point: A location in space with no size or dimension.

- Line: A straight one-dimensional figure having no thickness and extending infinitely in both directions.

- Line Segment: A part of a line that has two endpoints.

- Ray: A part of a line that has one endpoint and extends infinitely in one direction.

- Plane: A flat, two-dimensional surface that extends infinitely in all directions.

 2. Angles

- Acute Angle: An angle less than 90 degrees.

- Right Angle: An angle exactly 90 degrees.

- Obtuse Angle: An angle greater than 90 degrees but less than 180 degrees.

- Straight Angle: An angle exactly 180 degrees.

- Reflex Angle: An angle greater than 180 degrees but less than 360 degrees.

 3. Triangles

- Types of Triangles:

  - Equilateral Triangle: All sides and angles are equal (each angle is 60 degrees).

  - Isosceles Triangle: Two sides and two angles are equal.

  - Scalene Triangle: All sides and angles are of different measures.

  - Right-Angled Triangle: One angle is exactly 90 degrees.

- Properties of Triangles:

  - The sum of the interior angles is always 180 degrees.

  - The exterior angle is equal to the sum of the two opposite interior angles.

 4. Quadrilaterals

- Types of Quadrilaterals:

  - Square: All sides are equal, and all angles are 90 degrees.

  - Rectangle: Opposite sides are equal, and all angles are 90 degrees.

  - Rhombus: All sides are equal, but angles are not necessarily 90 degrees.

  - Parallelogram: Opposite sides are equal and parallel, and opposite angles are equal.

  - Trapezium: Only one pair of opposite sides is parallel.

  - Kite: Two pairs of adjacent sides are equal.

5. Circles

- Radius: The distance from the center to any point on the circle.

- Diameter: The distance across the circle passing through the center (twice the radius).

- Circumference: The perimeter of the circle (2πr).

- Chord: A line segment whose endpoints lie on the circle.

- Tangent: A line that touches the circle at exactly one point.

- Arc: A part of the circumference of the circle.

- Sector: A region bounded by two radii and an arc.

- Segment: A region bounded by a chord and an arc.

6. Polygons

- Regular Polygon: A polygon with all sides and all angles equal.

- Irregular Polygon: A polygon with sides and angles of different measures.

- Sum of Interior Angles: (n-2) × 180 degrees, where n is the number of sides.

7. Coordinate Geometry

- Cartesian Plane: A two-dimensional plane with a horizontal x-axis and a vertical y-axis.

- Distance Formula: The distance between two points (x₁, y₁) and (x₂, y₂) is √((x₂ - x₁)² + (y₂ - y₁)²).

- Midpoint Formula: The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).

- Slope of a Line: The slope (m) of a line passing through points (x₁, y₁) and (x₂, y₂) is (y₂ - y₁)/(x₂ - x₁).

8. Transformations

- Translation: Sliding a figure without rotating or flipping it.

- Reflection: Flipping a figure over a line (the line of reflection).

- Rotation: Turning a figure around a fixed point (the center of rotation).

- Dilation: Enlarging or reducing a figure by a scale factor from a fixed point.

9. Theorems and Postulates

- Pythagorean Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²).

- Angle Sum Property: The sum of the interior angles of a triangle is 180 degrees.

- Congruence and Similarity:

  - Congruent Figures: Figures that have the same shape and size.

  - Similar Figures: Figures that have the same shape but not necessarily the same size.

 10. Solid Geometry

- Cube: A three-dimensional shape with six square faces.

- Cuboid: A three-dimensional shape with six rectangular faces.

- Sphere: A perfectly round three-dimensional shape.

- Cylinder: A three-dimensional shape with two parallel circular bases connected by a curved surface.

- Cone: A three-dimensional shape with a circular base and a single vertex.

- Pyramid: A three-dimensional shape with a polygonal base and triangular faces that meet at a common vertex.

 11. Volume and Surface Area

- Cube:

  - Volume: V = a³ (where a is the length of a side).

  - Surface Area: SA = 6a².

- Cuboid:

  - Volume: V = l × w × h (where l is length, w is width, h is height).

  - Surface Area: SA = 2(lw + lh + wh).

- Sphere:

  - Volume: V = (4/3)πr³.

  - Surface Area: SA = 4πr².

- Cylinder:

  - Volume: V = πr²h.

  - Surface Area: SA = 2πr(h + r).

- Cone:

  - Volume: V = (1/3)πr²h.

  - Surface Area: SA = πr(r + l) (where l is the slant height).

- Pyramid:

  - Volume: V = (1/3)Bh (where B is the area of the base).

  - Surface Area: SA = B + (1/2)Pl (where P is the perimeter of the base and l is the slant height).