Geometry
Geometry is the branch of mathematics concerned with shapes, sizes, properties of space, and relative position of figures. It covers points, lines, angles, surfaces, and solids.
Basic Concepts
Some fundamental geometry terms:
- Point: A location in space with no size.
- Line: A collection of points extending infinitely in two directions.
- Plane: A flat, two-dimensional surface extending infinitely.
- Angle: Formed by two rays with a common endpoint called the vertex.
Types of Angles
- Acute angle: Less than 90°
- Right angle: Exactly 90°
- Obtuse angle: Between 90° and 180°
- Straight angle: Exactly 180°
- Reflex angle: Between 180° and 360°
Triangles
Triangles are three-sided polygons classified by side lengths or angles.
- Equilateral: All sides and angles equal (each 60°)
- Isosceles: Two sides and two angles equal
- Scalene: All sides and angles different
- Right triangle: One angle is 90°
Triangle Properties
Key properties and theorems:
- Sum\ of\ interior\ angles = 180^\circ
- Pythagorean\ theorem: a^2 + b^2 = c^2 (for right triangles)
- Triangle\ Inequality:\ The\ sum\ of\ any\ two\ sides\ >\ third\ side
Circles
A circle is a set of points equidistant from a center point.
- Radius (r): Distance from center to any point on circle
- Diameter (d): Twice the radius (d = 2r)
- Circumference (C): Perimeter of circle (C = 2\pi r)
- Area (A): Inside region (A = \pi r^2)
Example: Find the Area of a Triangle
Problem: Find the area of a triangle with base 10 cm and height 6 cm.
- Use formula: Area = \frac{1}{2} \times base \times height
- Area = \frac{1}{2} \times 10 \times 6 = 30
Answer: Area = 30 cm²
Practice Problems
1. Find the length of the hypotenuse of a right triangle with legs 3 cm and 4 cm.
- Use Pythagorean theorem: c^2 = a^2 + b^2
- c^2 = 3^2 + 4^2 = 9 + 16 = 25
- c = \sqrt{25} = 5
Answer: Hypotenuse = 5 cm
2. Calculate the circumference of a circle with radius 7 cm.
- Use formula: C = 2\pi r
- C = 2 \times \pi \times 7 \approx 43.98\ cm
Answer: Circumference ≈ 43.98 cm