# P and Q are the mid–points of the sides CA and CB respectively of ΔABC right angled at C. Prove that 4(AQ2 +BP2) = 5AB2

Given: ABC ia right triangle right angled at C

P and Q are the mid–points of the sides CA and CB respectively.

AP = PC and CQ = QB

In ACB, using Pythagoras Theorem, we have

(Perpendicular)2 + (Base)2 = (Hypotenuse)2

(AC)2 + (BC)2 = (AB)2 …(i)

Now, In ACQ, using Pythagoras Theorem, we have

(Perpendicular)2 + (Base)2 = (Hypotenuse)2

(AC)2 + (CQ)2 = (AQ)2

4(AC)2 + (BC)2 = 4(AQ)2

(BC)2 = 4(AQ)2 – 4(AC)2 …(ii)

Now, In PCB, using Pythagoras Theorem, we have

(Perpendicular)2 + (Base)2 = (Hypotenuse)2

(PC)2 + (BC)2 = (BP)2

(AC)2 + 4(BC)2 = 4(BP)2

(AC)2 = 4(BP)2 – 4(BC)2 …(ii)

Putting the value of (AC)2 and (BC)2 in eq. (i), we get

4(BP)2 – 4(BC)2 + 4(AQ)2 – 4(AC)2 = (AB)2

4(BP2 +AQ2) – 4(BC2 + AC2) = (AB)2

4(BP2 +AQ2) – 4(AB2) = (AB)2 [from eq(i)]

4(BP2 +AQ2) = 5(AB)2

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
Basic Proportionality Theorem42 mins
A Peep into Pythagoras Theorem43 mins
R.D Sharma | Solve Exercise -4.2 and 4.3FREE Class
NCERT | Strong Your Basics of Triangles39 mins
RD Sharma | Imp. Qs From Triangles41 mins
Quiz | Criterion of Similarity of Triangle45 mins
How to Ace Maths in NTSE 2020?36 mins
Know About Important Proofs in Triangles33 mins
Master BPT or Thales Theorem39 mins
Champ Quiz | Thales Theorem49 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
view all courses