Properties of Triangle Class 12 Mathematics Notes has been updated according to the latest syllabus of 2080. It means the solutions of Properties of Triangle chapter provided in this article contains all the new exercise that has recently been updated. Now you don’t need to go anywhere searching for the notes of this chapter because we are here to serve you.
Chapter – 6
Properties Of Triangle
A triangle consists of three sides and three angles. They are collectively known as the six elements of the triangle. Various relations between the sides and angles are known. There are relations connecting the area, sides, angles, circum-radius, ex-radius, in-radius etc.
Throughout our discussion, we denote the angles BAC, CBA, ACB of a triangle ABC by A, B, C respectively; and the lengths of the sides BC, CA, AB by a, b, c respectively. The circum-radius and the area of the triangle are denoted by R and A respectively.
The Sine Law
Let us consider a triangle ABC placed in the standard position with the vertex A at the origin and the side AB along the positive x-axis. The vertex of the triangle may be below or above the x-axis. If it is below the x-axis, the angle BAC, or simply A, will be negative; and if it is above the x-axis, it will be positive.
Properties of Triangle Class 12 Mathematics Notes PDF
This PDF will provide the solutions of every question from class 12 Properties of Triangle because this chapter has only one exercise .
The vertex C will lie on the first or fourth quadrant if A is acute; and on the second or third quadrant if it is obtuse. Denoting the sides of the triangle ABC opposite to the angles A, B and C by a, b and e respectively, we notice that
- the coordinates of A are (0, 0),
- the coordinates of B are (c, 0) and,
- the coordinates of C are (bcos A, bsin A) if C is above x-axis (bCos A, – bsin A).
Basic Properties of Triangles
- Sum of Interior Angles: The sum of the interior angles of a triangle is always 180 degrees.
- Exterior Angle Property: The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
- Triangle Inequality Theorem: In a triangle, the sum of the lengths of any two sides is greater than the length of the third side.
Special Properties of Triangles
- Equilateral Triangle: All sides are equal, and all angles are 60 degrees.
- Isosceles Triangle: Two sides are equal, and the base angles are equal.
- Scalene Triangle: No sides are equal, and no angles are equal.
- Right Triangle: One angle is 90 degrees, and it follows the Pythagorean theorem
Advanced Properties of Triangles:
- Median of a Triangle: A line segment joining a vertex to the midpoint of the opposite side.
- Altitude of a Triangle: A perpendicular line from a vertex to the opposite side.
- Orthocenter, Centroid, Circumcenter, and Incenter: Points of concurrency in a triangle with unique properties.
Understanding the properties of triangles is essential for solving geometric problems, proving theorems, and exploring the relationships between different elements within a triangle.
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