Q. 44.6( 7 Votes )

If in triangle ABC, AD is a median and AM is perpendicular to BC then prove that AB2= AD2 – BC × DM + 1/4 BC2.

Answer :

Given: D is the mid-point of BC.

And AM BC.



To Prove:


Proof: In right-angled ∆ABM,


By Pythagoras theorem, we can write as


AB2 = AM2 + BM2


[ (hypotenuse)2 = (perpendicular)2 + (base)2] …(i)


In right-angled ∆AMD,


By Pythagoras theorem, we can write as


AD2 = AM2 + MD2


[ (hypotenuse)2 = (perpendicular)2 + (base)2] …(ii)


Subtract equation (ii) from equation (i), we get


AB2 – AD2 = AM2 – AM2 + BM2 – MD2


AB2 – AD2 = 0 + BM2 – MD2


AB2 = AD2 – MD2 + BM2


AB2 = AD2 – DM2 + (BD – DM)2


AB2 = AD2 – DM2 + BD2 + DM2 – 2(BD)(DM)


AB2 = AD2 + BD2 – 2(BD)(DM)



[ ]



Hence, proved.


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Champ Quiz | Mastering TrianglesChamp Quiz | Mastering TrianglesChamp Quiz | Mastering Triangles44 mins
RD Sharma | Important Proofs in TrianglesRD Sharma | Important Proofs in TrianglesRD Sharma | Important Proofs in Triangles28 mins
R.D. Sharma | TrianglesR.D. Sharma | TrianglesR.D. Sharma | Triangles33 mins
NCERT | Most Important Questions of TrianglesNCERT | Most Important Questions of TrianglesNCERT | Most Important Questions of Triangles50 mins
Triangles - Important QuestionsTriangles - Important QuestionsTriangles - Important Questions39 mins
RD Sharma | Problems on TrianglesRD Sharma | Problems on TrianglesRD Sharma | Problems on Triangles41 mins
NCERT | Triangle InequalitiesNCERT | Triangle InequalitiesNCERT | Triangle Inequalities47 mins
NCERT | Practice Imp. Qs. on TrianglesNCERT | Practice Imp. Qs. on TrianglesNCERT | Practice Imp. Qs. on Triangles44 mins
Quiz | Are You Master of Triangles?Quiz | Are You Master of Triangles?Quiz | Are You Master of Triangles?48 mins
NCERT | Tricks to Apply Triangle InequalitiesNCERT | Tricks to Apply Triangle InequalitiesNCERT | Tricks to Apply Triangle Inequalities45 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses