Geometry: Understanding Triangles

Questions and Answers

What is the primary reason triangles are important in geometry?

They are used in trigonometry and the Pythagorean theorem. (correct)

They are the only shape with three sides.

They are the basis for circle geometry.

They are used in algebra and calculus.

What is the term for the point where two sides of a triangle meet?

Base

Side

Angle

Vertex (correct)

What is the relationship between the sides and angles of a triangle?

The sides are equal to the sum of the angles.

The angles are equal to the sum of the sides.

The largest angle is across from the largest side. (correct)

The smallest angle is across from the largest side.

What is the term for the horizontal side of a triangle?

Base Signup and view all the answers

How are the angles of a triangle typically labeled?

With uppercase letters Signup and view all the answers

What is a characteristic of a scalene triangle?

It has three unequal sides and three unequal angles Signup and view all the answers

What is the purpose of the Pythagorean theorem?

To show the relationship among the side lengths of a right triangle Signup and view all the answers

What is the formula for the area of a triangle given two sides and the included angle?

A = (a × b × sinC)/2 Signup and view all the answers

What is the purpose of the law of cosines?

To find the side lengths of a non-right triangle Signup and view all the answers

What is the unit of measurement often used in trigonometry for angles?

Radians Signup and view all the answers

What is the ratio of the circumference of a circle to the diameter of the circle?

π Signup and view all the answers

What is used to find a side of a right triangle when the other two sides are known?

Pythagorean Theorem Signup and view all the answers

What is used to find an angle of a right triangle?

SOH CAH TOA Signup and view all the answers

What is used to find an angle of a non-right triangle?

Law of Sines or Law of Cosines Signup and view all the answers

How many radians are in a circle?

2π Signup and view all the answers

Study Notes

Definition and Importance of Triangles

A triangle is a shape with three sides and three angles.
Triangles are crucial in geometry because they are used in many geometry proofs and are the basis for trigonometry and the Pythagorean theorem.

Triangle Properties

A triangle has three parts: sides, angles, and vertices.
Sides are line segments that form the triangle and are labeled with lowercase letters (e.g., a, b, and c).
Angles are formed by two sides and are labeled with uppercase letters (e.g., A, B, and C).
Vertices are the points where two sides meet and are labeled with uppercase letters.
The base of a triangle is one side, typically the horizontal side.
The height (or altitude) is the perpendicular distance from the base to the top vertex (or apex).

Types of Triangles

Triangles can be categorized based on sides or angles.
Types of triangles based on sides:
- Scalene triangle: three sides of different lengths.
- Isosceles triangle: two sides of equal length.
- Equilateral triangle: three sides of equal length.
Types of triangles based on angles:
- Obtuse triangle: one angle greater than 90°.
- Right triangle: one angle of 90°.
- Acute triangle: three angles less than 90°.

Triangle Formulas

Formulas for finding the area of a triangle:
- A = (b * h) / 2.
- A = (a * b * sin(C)) / 2.
- A = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter.
The Pythagorean theorem: a² + b² = c², where c is the hypotenuse.
The law of cosines: c² = a² + b² - 2ab * cos(C).
The law of sines: a / sin(A) = b / sin(B) = c / sin(C).

Trigonometry

Trigonometry is the study of relationships between angles and side ratios of triangles.
Basic trigonometric ratios: SOH CAH TOA.
Trigonometry formulas apply to all triangles, including right triangles.
Angles are often measured in radians, with 2π radians in a circle.

Examples of Triangle Formulas

Example 1: Finding the hypotenuse of a right triangle using the Pythagorean theorem.
Example 2: Finding an angle of a right triangle using SOH CAH TOA.
Example 3: Finding an angle of a non-right triangle using the law of cosines.