Flag 9 grade maths> Given that ABC is an isosceles right tria...
question mark

Given that ABC is an isosceles right triangle with AC=BC and angle ACB=90°.D is a point on AC and E is point on the extension of BD such that AE  is perpendicular to BE. If AE=1/2BD,prove that BD bisects angle ABC.

jsl ksl , 5 Years ago
Grade 9
anser 1 Answers
Aditya Gupta

Last Activity: 5 Years ago

let angle CBD= x. also, AC= BC= k
now, by angle chasing you can prove that angle DAE= angle CBD= x
now, BC= k= BDcosx.....(1)
and AE= ADcosx. but AD= AC – CD= k – BDsinx
so that AE= (k – BDsinx)cosx
or 2AE= BD= 2(k – BDsinx)cosx.........(2)
put k from (1) in (2)
BD= 2cosx(BDcosx – BDsinx)
or 1= 2cosx(cosx – sinx)
or 2cos^2x – 1= 2sinxcosx
or cos2x= sin2x
or tan2x= 1= tan45
or x= 22.5, which is obviously half of angle ABC (since angle ABC= 45 deg)
hence, BD bisects angle ABC.
kindly approve :)

Provide a better Answer & Earn Cool Goodies

Enter text here...
star
LIVE ONLINE CLASSES

Prepraring for the competition made easy just by live online class.

tv

Full Live Access

material

Study Material

removal

Live Doubts Solving

assignment

Daily Class Assignments


Ask a Doubt

Get your questions answered by the expert for free

Enter text here...