Triangle
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Triangles: Properties, Theorems, and Problem Solving - Prof. Cabug-Us, Lecture notes of Geometry
A comprehensive overview of triangles, including their properties, theorems, and various problem-solving techniques. It covers the fundamental characteristics of triangles, such as the sum of the three angles being equal to 180 degrees, the relationship between the sides and angles, and the concepts of similar and congruent triangles. The document also delves into specific theorems and formulas related to triangles, including the heron's formula, the cosine law, and the sine law, which are essential for calculating the area and other properties of triangles. Additionally, the document presents several sample problems with detailed solutions, demonstrating the application of these principles in real-world scenarios. This resource would be highly valuable for students studying geometry, mathematics, or related fields, as it offers a solid foundation for understanding and working with triangles.
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2022/2023
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Triangle
What is triangle?
A triangle is a polygon with three sides having
three vertices. The angle formed inside the
triangle is equal to 180 degrees.
Theorems and Properties
- If two sides of a triangle are equal (an isosceles triangle), the angles opposite these sides are equal.
- If two sides of a triangle are unequal, the angles opposite are unequal, and the greater angle is opposite the greater side
Theorems and Properties
- The perpendicular bisector of the sides, and the bisectors of the angles of a triangle, meet in points which are the center of the circumscribed circle and the inscribed circle, respectively.
Theorems and Properties
- The medians of a triangle are concurrent at a point which is two-thirds of the distance from the vertex to the midpoint of the opposite side. The point of concurrency is the centroid of the triangle.
Theorems and Properties
- If two triangles are similar, the ratio of their area is equal to the square of the ratio of their corresponding sides.
Note
- The term โRatio of the Perimeterโsโ is exactly the same as the similarity ratio (SR) and scale factor (SF). ๐๐น = ๐๐ =
2. (SF)
2 = (SR) 2 = ๐ด 1 ๐ด 2
Types of Triangles
Types of triangle according to its angle
Types of Triangles
Types of triangle according to both angle and sides
Formulas Relating to Triangle
Right Triangle
A = Area
๐๐ 2
2
2
2
Formulas Relating to Triangle
Oblique Triangle
Given three sides a, b, and c (Heronโs Formula) ๐ด = ๐(๐ โ ๐)(๐ โ ๐)(๐ โ ๐) ๐ =
Formulas Relating to Triangle
Oblique Triangle
Given three sides a, b, and c The area under this condition can also be solved by finding one angle using cosine law and apply the formula for two sides and included angle. ๐ด = ๐๐(๐ ๐๐๐ถ) 2 = ๐๐(๐ ๐๐๐ด) 2 = ๐๐(๐ ๐๐๐ต) 2 Cosine Law ๐๐๐ ๐ด = ๐ 2
- ๐ 2 โ ๐ 2 2๐๐ ๐๐๐ ๐ต = ๐ 2
- ๐ 2 โ ๐ 2 2๐๐ ๐๐๐ ๐ถ = ๐ 2
- ๐ 2 โ ๐ 2 2๐๐
Sample Problem 1
The corresponding sides of two similar
triangles has similarity ratio 3:2. What is the
ratio of their รกreas?
Solution
If two triangles are similar, the ratio of their
area is equal to the square of the ratio of their
corresponding sides
1
2
2
2