त्रिभुज(About area of triangle)
A plane closed figure bounded by three line segments is called a triangle. The triangle is denoted by . A triangle has six parts – three sides and three angles.
Area of triangle if height and base are known
Heron’s Formula(About area of triangle)
Formula is Where is the semiperimeter of the triangle and , And The lengths of the sides of the triangle are
Here S=(a+b+c)/2 and a, b and c are the three sides of the triangle.
Types of Triangles
Triangles are divided on two bases:
(i) on the basis of angles
(ii) at the base of the sides
(i) Types of Triangles on the Basis of Angles
Right-angled triangle :
A triangle whose measure of one angle is 90° is called a right angled triangle. The side opposite to the right angle is called the hypotenuse of the triangle, the remaining two sides are called the base and the perpendicular respectively.
(hypotenuse)^2 = (base)^2 + (perpendicular)^2
Acute-angled triangle :
A triangle whose measure of each angle is greater than 0° and less than 90° is called an acute triangle.
Obtuse-angled triangle :
A triangle whose measure of one angle is greater than 90° but less than 180° is called an obtuse triangle.
(ii) Types of Triangle on the Basis of Sides
Scalene triangle :
A triangle whose all the sides are of different lengths is called an equilateral triangle.
A triangle whose two sides are equal is called an isosceles triangle.
A triangle whose three sides are equal is called an equilateral triangle.
Area of Equilateral triangle = ,where s=length of the side
Some Useful Results on Triangles
• The sum of the three angles of a triangle is 180°.
• The measure of the exterior angle formed by extending one side of a triangle is equal to the sum of the measures of two opposite interior angles.
• The exterior angle of a triangle is greater than any opposite interior angle.
• The angle between the interior of one base angle of a triangle and the exterior angle of another angle is half of the angle.
• The sum of any two sides of a triangle is greater than its third side.
•If the bisector of the vertex angle of a triangle bisects the base, then the triangle is an isosceles triangle.
• The perpendiculars from the vertices of an isosceles triangle on opposite sides are equal.
•The medians of an equilateral triangle are equal.
Congruent Triangle :
If all the angles and sides of one triangle are equal to the corresponding angles and corresponding sides of another triangle, then both the triangles are said to be congruent.
• Every triangle is congruent to itself.
Triangle Congruency Axiom
Assumption 1: If two sides of one triangle and the angle between them are equal to the corresponding sides and corresponding angle of the other triangle. (SAS axiom)
Assumption 2: If two angles of one triangle and the common side of these angles are equal to the corresponding angles and common side of the other triangle. (SAA axiom)
Assumption 3: If all the three sides of one triangle are equal to the three corresponding sides of another triangle.(SSS axiom)
Assumption 4: If the hypotenuse and one side of one triangle are equal to the hypotenuse and the corresponding side of the other triangle. (RHS axiom)
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