Classification Of Triangles

Triangles can be classified based on the lengths of their sides and the measures of their angles.

By Sides

This section classifies triangles based on their sides and angles.

By Angles

This section discusses the classification of triangles based on their angles, introducing concepts such as acute, right, and obtuse triangles.

Triangle Inequality Theorem

The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Angles Of A Triangle

The section focuses on the fundamental properties of angles within a triangle, including their sum and the exterior angle theorem.

Sum Of Angles

The section covers the fundamental property of triangles where the sum of the interior angles equals 180°, including the application of the exterior angle theorem.

Exterior Angle Theorem

The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Congruence (≅)

Congruence in triangles refers to the condition where two triangles are equal in all corresponding sides and angles.

Key Criteria

This section covers the classification of triangles, including side and angle classifications, the Triangle Inequality Theorem, angle properties, triangle congruence and similarity criteria, special centers, and relevant trigonometric concepts.

Applications

This section explores various applications of triangular properties in real-world contexts such as engineering, physics, and architecture.

Similarity (∼)

This section discusses triangle similarity, highlighting the relationships between corresponding angles and sides.

Criteria

This section covers the classification, properties, and theorems related to triangles, including congruence and similarity.

Properties

This section covers the properties of triangles, including classifications, the triangle inequality theorem, angle relationships, congruence, similarity, and special centers.

Special Centers Of A Triangle

This section explores the special centers of a triangle, including the centroid, incenter, circumcenter, and orthocenter, highlighting their properties and significance.

Centroid (G)

The centroid of a triangle, denoted as G, is the point where the three medians of the triangle intersect, effectively dividing each median into segments with a 2:1 ratio from the vertex.

Incenter (I)

The incenter of a triangle is the point where the angle bisectors intersect, equidistant from all sides and serving as the center of the inscribed circle.

Circumcenter (O)

The circumcenter is the point where the perpendicular bisectors of a triangle intersect, and it serves as the center of the triangle's circumcircle, being equidistant from all three vertices.

Orthocenter (H)

The orthocenter (H) of a triangle is the intersection point of its altitudes, and its position varies based on the type of triangle.

Pythagorean Theorem & Its Converse

The Pythagorean Theorem establishes the relationship between the sides of a right triangle, while its converse clarifies how to identify right triangles.

Trigonometry In Right Triangles

This section covers trigonometric ratios for right triangles, including sine, cosine, and tangent.

Definitions

This section introduces key definitions and properties related to triangles, including classifications, angle relationships, and special centers.

Basic Identities

This section introduces fundamental trigonometric identities, allowing students to relate sine, cosine, and tangent functions.

Worked Example

This worked example illustrates how to apply trigonometric ratios in a right triangle to find unknown side lengths and angles.

Area Of A Triangle

The section discusses various methods to calculate the area of a triangle including the basic formula, trigonometric formulas, and Heron's formula.

Basic Formula

This section introduces the basic formulas for calculating the area of a triangle and highlights different methods, including Heron's formula.

Trigonometric Formula

This section introduces the trigonometric formula for calculating the area of a triangle using the sine of an angle between two sides.

Heron’s Formula

Heron's formula allows for the calculation of the area of a triangle when the lengths of all three sides are known.

Example (Heron)

Law Of Sines & Law Of Cosines

The Law of Sines and Law of Cosines are crucial for solving non-right triangles, particularly in calculating unknown sides and angles.

Law Of Sines

The Law of Sines establishes a relationship between the angles and sides of a triangle, enabling the solution of non-right triangles.

Law Of Cosines

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles, useful for solving any type of triangle.

Useful Applications

This section discusses the practical applications of the Law of Sines and the Law of Cosines in solving triangle-related problems.

Detailed Worked Examples

This section presents detailed worked examples utilizing the Law of Sines and Law of Cosines, showcasing practical applications of triangle properties.

Summary

The section summarizes key properties of triangles, including classification, congruence, similarity, special centers, and their applications in problem-solving.

Final Thoughts

Mastering the properties of triangles is crucial for advancing in geometry and trigonometry.