Answer :

Given: ABC is an isosceles triangle. So, AC = BC


C = right angle



To prove: AD2 + DB2 = 2CD2


Proof:
By Pythagoras Theroem, 

In ΔABC, We have

AB2 = AC2 + BC2                 [1]

In ΔADC and ΔADB, we have

AC = BC               [Given]

CD = CD              [Common]

AD = DB              [D is the mid point]

⇒ ΔADC ≅ ΔADB             [By SSS Congruency Criterion]

⇒ ∠ADC = ∠ADB           [Corresponding parts of congruent triangles are equal]

Also, ∠ADC + ∠ADB = 180°      [Linear Pair]

⇒ ∠ADC + ∠ADC = 180°

⇒ ∠ADC = ∠ADB = 90°

Hence, ADC and ADB are right triangles, ∴ By Pythagoras theorem

AC2 = CD2 + AD2            [2]

BC2 = CD2 + BD2            [3]

 Adding [2] and [3], we have

AC2 + BC2 = AD2 + BD2 + 2CD2

⇒ AB2 = AD2 + BD2 + 2CD2

⇒ (2AD)2 = AD2 + AD2 + 2CD2              

[∵ D is mid-point of AB, AD = BD and AB = 2AD = 2BD]

⇒ 4AD2 = 2AD2 + 2CD2

⇒ 2AD2 = 2CD2

⇒ AD2 + BD2 = 2CD2    [∵ AD = BD]

Hence, Proved!

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
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
RELATED QUESTIONS :

A person goes 24mWest Bengal Board - Mathematics

Two rods of 13m. West Bengal Board - Mathematics

In ΔABC, if AB = West Bengal Board - Mathematics

If the lengths ofWest Bengal Board - Mathematics

In the adjoining West Bengal Board - Mathematics

I have drawn a trWest Bengal Board - Mathematics

Let us fill in blWest Bengal Board - Mathematics

In ΔRST, <span laWest Bengal Board - Mathematics

In the triangle AWest Bengal Board - Mathematics

In the road of ouWest Bengal Board - Mathematics