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Types of Triangle and Their Definition

Written By satyam coaching centre on Sunday, 15 September 2019 | 08:22

Types of Triangle and Their Definition-

1.The figure formed by connecting three noncollinear points (the vertices) by line segments.

(2.)The figure described in (1)together with the points in the same plane and interior to the figure. Six kinds of triangles are illustrated. As indicated in the above figures, an acute triangle is a triangle whose interior angles are all acute ;an obtuse triangle is a triangle that contains an obtuse interior angle ;a scalene triangle is a triangle is a triangle with two equal sides (the third side is called the base and the angle opposite it the vertex) ;a right triangle is a triangle one of whose angles is a right angle (the side opposite the right angle is called the hypotenuse and the other two sides the legs of the right triangle) ;an equilateral triangle is a triangle with all three sides equal (it must then also be equiangular, i. e., have its three interior angles equal). An oblique triangle is a triangle which contains no right angles. The altitude of a triangle is the perpendicular distance from a vertex to the opposite side, which has been designed as the base. The area of a triangle is one-half the product of base and the corresponding altitude. The area is equal to one-half the determinant whose first column consists of the abscissas of the vertices, the second of the ordinates (in the same order), and the third entirely of ones (this is positive if the points are taken around the triangle in counterclockwise order.)
Astronomical triangle - The spherical triangle on the celestial sphere which has for its vertices the nearer celestial pole, the zenith, and the celestial body under consideration.

2.Congruent triangles-

Types of Triangle and Their Definition

Types of Triangle and Their Definition

In Plane geometry, it is customary to say that two figures are congruent if one of them can be made to coincide with the other by a rigid motion in space (i.e., by translations and rotations in space). Thus it might be said that two figures are congruent if they "differ only length in location."Two line segments of equal length are congruent and two circles of equal radii are congruent. Each of the following is a necessary and sufficient condition for two triangles to be congruent:(i.) There is a one-to-one correspondence between the sides of the other for which corresponding sides are equal (ii.) there is a one-to-one correspondence between the sides of one triangle and the sides of the other for which two sides and the angle, determined by these sides are equal, respectively, to the corresponding sides of the other triangle and the angle determined by these sides (SAS) ;(iii) there is a one-, to - one correspondence between the angles of one triangle and the angles of the other for which two angles and the sides between the vertices of these angles are equal, respectively, to the corresponding angles of the other triangle and the side between the vertices of these angles (ASA). If we change the definition of congruence to allow only rigid motions in the plane, a different concept of congruence results. In solid geometry, two figures are congruent if one of them can be made to coincide with the other by a rigid motion in space. Sometimes such figures are said to be directly congruent and two figures for which one is directly congruent to the reflection of the other through a plane are oppositely congruent (then two figures are either directly or oppositely congruent if and only if one can be made to coincide with the other by a rigid motion in four-dimensional space). Often when giving axioms for a geometric system,congruence is taken as an undefined concept restricted by suitable axioms.


3.Pascal's triangle -

A triangular array of numbers composed of the coefficients in the expansion composed of the coefficients in the expansion of (x+y) ^n for n=0,1,2,3,etc.The "triangle" extends down indefinitely, the coefficients in the expansion of (x+y) ^n being in the (n+1)st row. As shown, the array is bordered by 1's and the sum of two adjacent numbers in one row is equal to the number in the next row between the two numbers. The array is symmetric about the vertical line through the "vertex"
                                         1
                                  1               1
                             1           2            1
                        1         3             3           1
                    1        4           6           4          1
               1         5          10         10         5     1

4.Pedal Triangle -

The triangle formed within a given triangle by joining the feet of the perpendiculars from any given point to the sides. The triangle DEF is the pedal triangle formed within the triangle ABC by the joining the feet of the altitude. The figure illustrates the fact that the altitudes of the given triangle bisect the angles of this pedal triangle.

5.Polar Triangle -

Polar triangle of a spherical triangle. The a spherical triangle whose view is ertices are poles of the sides of the given triangle, the poles being the ones nearest to the vertices opposite the sides of which they are poles.

6.Solution of a Triangle -

Finding the remaining angles and sides when sufficient of these have been given For a plane right triangle, it is sufficient to know any two sides, or to know one of the acute angles and one side. The unknown parts are found by use of trigonometric tables and the definitions of the trigonometric functions :if a, b, c represent the legs and hypotenuse, respectively, and A, B are the angles opposite sides a and b then a=b then a=b tan A =c sin A, b=c cosA, A =tan^-1(a/b),B=90-A. For an oblique plane triangle, it is sufficient to know all three sides and one angle (, except that when two sides and the angle opposite one of them is given there may be two solutions.
For a right spherical triangle, Napier's rules supply all the formulas needed. For formulas providing solutions of an oblique spherical triangle in cases when solutions of an oblique spherical triangle in cases when solutions exist.

7.Spherical Triangle -

A spherical polygon with three sides ;a portion of a sphere bounded by three arcs of great circles. In the spherical triangle ABC, the sides of the triangle are a=angle BOC, b=angle AOC, and c=angle AOB. The angles of the triangle. 

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