The Triangle And Its Properties | Study

Triangle: 

A closed plane figure bounded by three line segments is called Triangle. 

The six elements of a triangle are its three angles and the three sides. The line segment joining a vertex of a triangle to the midpoint of its 

Median: 

The opposite side is called the median of the triangle.

A triangle has three medians. 

Altitude of the triangle:

The perpendicular line segment from vertex of a triangle to its opposite sides is called an altitude of the triangle. 

A triangle has3 altitudes.

Type of triangle based onSides: Equilateral: 

A triangle is said to be equilateral, if each one of its sides has the same length. In An equilateral triangle, each angle measures 60°. 

Isosceles Triangle: 

A triangle is said to be isosceles, if atleast any two of its sides are of same length. The non-equal side of an isosceles triangle is called its base; the base angles of an isosceles triangle have equal measure. 

Scalene Triangle: 

A triangle having all sides of different lengths. It has no two angles equal. 

Property of the lengths of sides of a triangle: 

The sum of the lengths of any two sides of a triangle is greater than the length of the third side. The difference between the lengths of any two sides is smaller than the length of the third side. This property is useful to know if it is possible to draw a triangle when the lengths of the three sides are known. 

Types of Triangle based on Angles: 

(i) Right Angled Triangle: 

A triangle one of whose angles measures 

(ii) Obtuse Angled Triangle: 

A triangle one of whose angles measures more than 

(iii) Acute Angled Triangle: 

A triangle each of whose angles measures less than In a right angled triangle, the side opposite to the right angle is called the hypotenuse and the other two sides are called its legs. 

Pythagoras property: 

In a right-angled triangle, the square on the hypotenuse = the sum of the squares on its legs.If a triangle is not right-angled, this property does not hold good. Thisproperty is useful to decide whether a given triangle is right-angled

or not. 

Exterior angle of a triangle:

An exterior angle of a triangle is formed, when a side of a triangle is produced. At each vertex, you have two ways of forming an exterior angle. 

A property of exterior angles: 

The measure of any exterior angle of a triangle is equal to the sum of the measures of its interior opposite angles. 

The angle sum property of a triangle:

 The total measure of the three angles of a triangle is 180°.

Property of the Lengths of Sides of a Triangle: 

The sum of the lengths of any two sides of a triangle is always greater than the length of the third side. The difference of the lengths of any two sides of a triangle is always smaller than the length of the third side.

Important Formulas – TheTriangles and its Properties 

1. A triangle is a figure made up by three line segments joining, in pairs, three non-collinear points. That is, if A, B, C are three non-collinear points, the figure formed by three line segments AB,BC and CA is called a triangle with vertices A, B, C. 

2. The three line segments forming a triangle are called the sides of the triangle. 

3. The three sides and three angles of a triangle are together called the six parts or elements of the triangle. 

4. A triangle whose two sides are equal, is called an isosceles triangle. 

5. A triangle whose all sides are equal, is called an equilateral triangle. 

6. A triangle whose no two sides are equal, is called a scalene triangle. 

7. A triangle whose all the angles are acute is called an acute triangle. 

8. A triangle whose one of the angles is a right angle is called a right triangle. 

9. A triangle whose one of the angles is an obtuse angle is called an obtuse triangle. 

10. The interior of a triangle is made up of all such points P of the plane, as are enclosed by the triangle. 

11. The exterior of a triangle is that part of the plane which consists of those points Q, which are neither on the triangle nor in its interior. 

12. The interior of a triangle together with the triangle itself is called the triangular region. 

13. The sum of the angles of a triangle is two right angles or 180°. 

14. If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the interior opposite angles.

 15. In any triangle, an exterior angle is greater than either of the interior opposite angles. 

16. The sum of any two sides of a triangle is greater than the third side. 

17. In a right triangle, if a, b are the lengths of the sides and c that of the hypotenuse, then

18. If the sides of a triangle are of lengths a, b and c such that

then the triangle is right-angled and the side of length c is the hypotenuse. 

19. Three positive numbers a, b, c in this order are said to form a Pythagorean triplet, if

Triplets (3, 4, 5) (5, 12,13), (8, 15, 17), (7,24, 25) and (12, 35,37) are somePythagorean triples.