PROPERTIES OF
TRIANGLES
Parts of a
triangle
We know that a triangle has six parts viz.
Three angles and three sides.
In any triangle ABC, the measures of the
angles BAC, CBA and ACB are denoted respectively by the letters A, B and C, and
the sides BC,CA and AB opposite to these angles are represented by a, b and c
respectively. The radius of the circumcircle, i.e. Circumradius will be denoted
by R and the area of the triangle of . All
the six parts of the triangle are not independent; they satisfy certain
relations.
In geometry you are familier with the
following relations :
(i)
A + B + C = 180 degree (ii) a+b > c b + c > a c + a > b
Law of sines or
sine rule
a/sin
A = b/sin B = c/sin C
Law of cosine
or cosine rule
1.
a^2 = b^2 + c^2 -2bc cosA or cos
A = (b^2 + c^2 – a^2) / 2bc
2.
b^2 = c^2 + a^2 – 2ca cos B or cos
B = (c^2 + a^2 – b^2) / 2ca
3.
c^2 = a^2 + b^2 – 2ab cos C or cos
C = (a^2 + b^2 – c^2) / 2ab
Law of tangents
or tangent rule
1.
tan (B-C)/2 = (b-c)/(b+c) cot A/2
2.
tan (C-A)/2 = (c-a)/(c+a) cot B/2
3.
tan (A-B)/2 = (a-b)/(a+b) cot C/2
The relation
between the three sides and two angles of any triangle ABC are as follows
1.
a = b cos C + c cos B
2.
b = c cos A +a cos C
3.
c = a cos B + b cos A
To find sines of half an angle of a
triangle in terms of the sides
1.
sin A/2 = (s-b)(s-c)/bc
2.
sin B/2 = (s-c)(s-a)/ca
3.
sin C/2 = (s-a)(s-b)/ab
To find cosines
of half an angle of a triangle in terms of the sides
1.
cos A/2 = s(s-a)/bc
2.
cos B/2 = s(s-b)/ca
3.
cos C/2 = s(s-c)/ab
To find tangent
of half an angle of triangle in terms of the sides
1.
tan A/2 = (s-b)(s-c) / s(s-a)
2.
tan B/2 = (s-c)(s-a) / s(s-b)
3.
tan C/2 = (s-a)(s-b) / s(s-c)
Expressions for
area of a triangle ABC
1.
Area = 1/2 ab sin C
2.
Area = 1/2 bc sin A
3.
Area = 1/2 ca sin B
Values of sines of angles in terms of sides and area
1.
sin A = 2/bc s(s-a)(s-b)(s-c)
2.
sin B = 2/ca s(s-a)(s-b)(s-c)
3.
sin C = 2/ab s(s-a)(s-b)(s-c)
Values of
tangents and cotangents of half an angle in terms of sides and area
1.
tan A/2 = (s-b)(s-c) / s(s-a)(s-b)(s-c)
2.
tan B/2 = (s-c)(s-a) / s(s-a)(s-b)(s-c)
3.
tan C/2 = (s-a)(s-b) / s(s-a)(s-b)(s-c)
4.
cot A/2 = s(s-a) / s(s-a)(s-b)(s-c)
5.
cot B/2 = s(s-b) / s(s-a)(s-b)(s-c)
6.
cot C/2 = s(s-c) / s(s-a)(s-b)(s-c)
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