Internal bisector of angle A of a triangle ABC meets the side BC at D.
A line drawn through D perpendicular to AD intersects the AC at E
and the side AB at F. If a,b,c represent the sides of triangle ABC, then
 
1)AE is the H. M.of b and 2c.     2)AD=2bc/b+c cos^2 A/2
 
3)EF=4bc/b+c sin A/2.                4)None of these

Asked by Sunil Soni | 7th Dec, 2015, 11:03: PM

Expert Answer:

A r e a space o f space increment A B C space equals space A r e a space o f space increment A B D space plus space A r e a space o f space increment A D C rightwards double arrow 1 half b c sin A space equals space 1 half c A D sin A over 2 plus 1 half b A D sin A over 2 rightwards double arrow 1 half b c cross times 2 sin A over 2 cos A over 2 equals 1 half A D sin A over 2 open parentheses c plus b close parentheses rightwards double arrow 2 b c cos A over 2 equals A D open parentheses b plus c close parentheses rightwards double arrow A D equals fraction numerator 2 b c over denominator b plus c end fraction cos A over 2... left parenthesis 1 right parenthesis  I n space r i g h t space t r i a n g l e space A D E comma A E space equals space fraction numerator A D over denominator cos begin display style A over 2 end style end fraction equals space fraction numerator fraction numerator 2 b c over denominator b plus c end fraction c o s A over 2 over denominator cos begin display style A over 2 end style end fraction rightwards double arrow A E equals fraction numerator 2 b c over denominator b plus c end fraction... left parenthesis 2 right parenthesis E F space equals space E D space plus space D F equals space A D tan A over 2 plus A D t a n A over 2 equals 2 A D tan A over 2 equals 2 cross times fraction numerator 2 b c over denominator b plus c end fraction c o s A over 2 cross times t a n A over 2 equals fraction numerator 4 b c over denominator b plus c end fraction sin A over 2.... left parenthesis 3 right parenthesis F r o m space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis space a n d space left parenthesis 3 right parenthesis comma space o p t i o n space left parenthesis 3 right parenthesis space i s space c o r r e c t.

Answered by satyajit samal | 8th Dec, 2015, 12:27: PM