Comparing Quantities

Introduction to Triangles

A triangle is the most basic closed geometric shape, formed by three line segments joining three points called vertices. Each pair of vertices is connected by a side, resulting in three sides and three vertices. The three sides meeting at the vertices create three angles inside the triangle, known as the angles of the triangle. Triangles come in various shapes and sizes, such as equilateral, isosceles, scalene, right-angled, acute-angled, and obtuse-angled triangles. The notation for a triangle is typically Δ followed by the vertices, for example, ΔABC. The angles at the vertices are denoted as ∠A, ∠B, and ∠C, corresponding to vertices A, B, and C respectively. The naming order of vertices can vary without changing the triangle's identity. Understanding triangles is fundamental in geometry, as they form the basis of many geometric principles and constructions.

• A triangle consists of three vertices and three sides.
• The sides are line segments joining pairs of vertices.
• The three sides form three interior angles at the vertices.
• Triangles are denoted by Δ followed by the vertices, e.g., ΔABC.
• Angles at vertices are denoted as ∠A, ∠B, and ∠C.
• Triangles can be classified based on side lengths and angle measures.
• 📌 Triangle: A closed figure with three sides and three vertices.
• 📌 Vertex: A corner point of a triangle.
• 📌 Side: A line segment joining two vertices.

7.1 Equilateral Triangles

Equilateral triangles are those in which all three sides are equal in length, making them highly symmetric. Constructing an equilateral triangle with a given side length, such as 4 cm, can be done using a compass and ruler. Simply drawing the sides with a ruler may require multiple attempts to ensure all sides are equal. Using a compass simplifies this process: after drawing the base AB of length 4 cm, draw arcs of radius 4 cm centered at points A and B. The intersection of these arcs marks the third vertex C, ensuring AC = BC = AB = 4 cm. Joining these points forms the equilateral triangle. This method guarantees precision and efficiency in construction.

• Equilateral triangles have all sides equal.
• Constructing with just a ruler may require multiple trials.
• Using a compass allows precise construction by drawing arcs.
• The intersection of arcs from two vertices determines the third vertex.
• All sides and angles in an equilateral triangle are equal.
• This construction method is efficient and accurate.
• 📌 Equilateral triangle: A triangle with all sides equal.
• 📌 Compass: A tool used to draw arcs and circles with a fixed radius.
• 📌 Arc: A portion of a circle's circumference.