Questions and Answers
What is the primary reason triangles are important in geometry?
What is the term for the point where two sides of a triangle meet?
What is the relationship between the sides and angles of a triangle?
What is the term for the horizontal side of a triangle?
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How are the angles of a triangle typically labeled?
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What is a characteristic of a scalene triangle?
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What is the purpose of the Pythagorean theorem?
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What is the formula for the area of a triangle given two sides and the included angle?
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What is the purpose of the law of cosines?
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What is the unit of measurement often used in trigonometry for angles?
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What is the ratio of the circumference of a circle to the diameter of the circle?
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What is used to find a side of a right triangle when the other two sides are known?
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What is used to find an angle of a right triangle?
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What is used to find an angle of a non-right triangle?
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How many radians are in a circle?
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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.
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Description
Learn about triangles in geometry, their definition, importance, and properties. Discover how they are used in proofs and problem-solving contexts.