Triangle Basics and Properties

Study Notes

Triangle

Definition
- A triangle is a polygon with three edges and three vertices.
Types of Triangles
- By sides:
  - Equilateral: All sides are equal; all angles are 60°.
  - Isosceles: Two sides are equal; two angles are equal.
  - Scalene: All sides and angles are different.
- By angles:
  - Acute: All angles are less than 90°.
  - Right: One angle is exactly 90°.
  - Obtuse: One angle is greater than 90°.
Properties of Triangles
- The sum of interior angles is always 180°.
- The longest side is opposite the largest angle.
- The shortest side is opposite the smallest angle.
Triangle Inequality Theorem
- The sum of the lengths of any two sides must be greater than the length of the third side.
Area Calculation
- Base and Height: Area = (1/2) × base × height
- Heron's Formula: Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 and a, b, c are the sides.
Perimeter Calculation
- Perimeter = a + b + c, where a, b, and c are the lengths of the sides.
Special Triangles
- 45-45-90 Triangle: Sides are in the ratio 1:1:√2.
- 30-60-90 Triangle: Sides are in the ratio 1:√3:2.
Congruence Criteria
- SSS: Side-Side-Side
- SAS: Side-Angle-Side
- ASA: Angle-Side-Angle
- AAS: Angle-Angle-Side
- HL: Hypotenuse-Leg (for right triangles)
Similarity Criteria
- AA: Angle-Angle
- SSS: Side-Side-Side (proportional sides)
- SAS: Side-Angle-Side (one angle and proportional sides)
Coordinate Geometry
- Triangle vertices can be represented by coordinates (x1, y1), (x2, y2), (x3, y3).
- Area can be calculated using the formula: Area = (1/2) | x1(y2-y3) + x2(y3-y1) + x3(y1-y2) |.
Triangle Centers
- Centroid: Intersection of medians; average of vertex coordinates.
- Orthocenter: Intersection of altitudes.
- Circumcenter: Intersection of perpendicular bisectors; center of circumcircle.
- Incenter: Intersection of angle bisectors; center of incircle.

Triangle Overview

A triangle is a geometric shape defined as a polygon with three edges and three vertices.

Types of Triangles

By Sides:
- Equilateral Triangle: All three sides and angles are equal (each angle measures 60°).
- Isosceles Triangle: Two sides are equal, leading to two equal angles opposite these sides.
- Scalene Triangle: All sides and angles are different from each other.
By Angles:
- Acute Triangle: All internal angles are less than 90°.
- Right Triangle: Contains one angle that is exactly 90°.
- Obtuse Triangle: Includes one angle that is greater than 90°.

Properties of Triangles

The interior angles of a triangle always sum to 180°.
The longest side is opposite the largest angle, while the shortest side is opposite the smallest angle.

Triangle Inequality Theorem

For any triangle, the sum of the lengths of any two sides must exceed the length of the third side.

Area Calculation

Using Base and Height: Area is calculated as (1/2) × base × height.
Heron’s Formula: Area can also be computed using the formula Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 and a, b, c are the lengths of the sides.

Perimeter Calculation

The perimeter of a triangle is the sum of the lengths of all sides: Perimeter = a + b + c.

Special Triangles

45-45-90 Triangle: The sides are in the ratio of 1:1:√2.
30-60-90 Triangle: The sides are in the ratio of 1:√3:2.

Congruence Criteria

SSS: Side-Side-Side (three pairs of equal sides).
SAS: Side-Angle-Side (two sides and the included angle are equal).
ASA: Angle-Side-Angle (two angles and the included side are equal).
AAS: Angle-Angle-Side (two angles and a non-included side are equal).
HL: Hypotenuse-Leg (applicable for right triangles).

Similarity Criteria

AA: Angle-Angle (two pairs of equal angles).
SSS: Side-Side-Side (proportional sides).
SAS: Side-Angle-Side (one angle equal and proportional sides).

Coordinate Geometry

The vertices of a triangle can be represented by coordinates (x1, y1), (x2, y2), (x3, y3).
The area can be determined with the formula: Area = (1/2) | x1(y2-y3) + x2(y3-y1) + x3(y1-y2) |.

Triangle Centers

Centroid: The point where the three medians intersect, calculated as the average of the vertex coordinates.
Orthocenter: The intersection point of the triangle's altitudes.
Circumcenter: The intersection of the perpendicular bisectors; this point is the center of the circumcircle.
Incenter: The location where the angle bisectors of the triangle meet; it serves as the center of the incircle.

Description

This quiz covers the fundamental concepts of triangles, including their definitions, types, properties, and important theorems. Test your understanding of triangle area and perimeter calculations with various methods. Perfect for students and math enthusiasts alike!