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subject: Solve Fact Triangles [print this page]

Introduction for triangles: A fact triangle is one of a basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A fact triangle with vertices A, B, and C is denoted ABC. In Euclidean geometry any three non-collinear points determine unique fact triangle and a unique plane.

Example Problems for Fact Triangles:

Ex 1: Solve any ?XYZ, prove and the question:

a3 sin(Y -Z) +b^3 sin( Z -X) +c3 sin(X -Y)= 0

Sol : By the sine rule, we have

a / sin X= b / sin Y = Z / sin Z =k(say)

=> a = ksin X ,b=Ksin Y and c = Ksin Z.

a^3 sin(X - Z)= K^3 sin^2 X[sin X sin(Y -Z)]

= K^3 sin^2 X [sin(? - (Y + Z)} sin(Y -Z)] in above equation to solve

= K^3 sin^2 X [sin(Y + Z)sin(Y - Z)]

= K^3 sin^2 X. (sin^2 Y - sin^2 Z).

Similarly to solve , b^3 sin(Z - X)=K^3 sin^2 Y(sin^2 Z - sin^2 X)

And, c^3 sin(X -Y)= K^3 sin^2 Z(sin^2 X - sin^2 Y)

LHS= K^3 sin^2 X(sin^2 Y - sin^2 Z) + K^3 sin^2 Y(sin^2 Z - Sin2X) + K^3 sin^2 Z(sin^2 X -sin^2 Y)

=0 = RHS.

Q 2 : Solve any ?ABC, and solve by using fact triangles:

(a -b)2 cos^2 C /2 +(a + b)2 sin^2 C / 2 =c2

Sol : We have

LHS = (a -b)2 cos^2 C /2 +(a +b)2 sin^2 C /2

= a^2(cos^2 C /2 + sin^2 C /2) + b2(cos^2 C /2 +sin^2 c/2) -2ab cos^2 C /2 +2absin^2 c /2

= a^2 + b2 - 2ab(cos^2 C /2 - sin^2 c /2)

= a^2 + b2 - 2ab.(a^2 + b2 -c2) / 2ab = c2 =RHS

Practice Problem for Fact Triangles

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Q 1: Solve any ?ABC, solve the question

2(asin^2 c / 2 + csin^2 A / 2)=(a + c -

Ansr: (a+ c - b) = RHS

This is a practice problem for fact triangles.

Q 2 : Solve any ?ABC, solve the question:

a^3 cos(B -C) + b^3 cos( C - A) +c^3 cos( A -b) = 3abc

Ans : (abc + abc + abc) = 3abc = RHS

This is a practice problem for fact triangles.

by: johnharmer

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