**
Question 19 (OR 2
**
**
nd
**
**
Question)
**

In ∆ABC, if AD is the median, then show that AB
^{
2
}
+ AC
^{
2
}
= 2(AD
^{
2
}
+ BD
^{
2
}
)

Last updated at Nov. 1, 2018 by Teachoo

**
Question 19 (OR 2
**
**
nd
**
**
Question)
**

In ∆ABC, if AD is the median, then show that AB
^{
2
}
+ AC
^{
2
}
= 2(AD
^{
2
}
+ BD
^{
2
}
)

Transcript

Question 19 (OR 2nd Question) In ∆ABC, if AD is the median, then show that AB2 + AC2 = 2(AD2 + BD2) Given: AD is a Median of ∆ABC ∴ BD = CD Also, AE ⊥ BC To Prove: AB2 + AC2 = 2(AD2 + BD2) Proof : In Right ∆ABE Applying Pythagoras Theorem, AB2 = AE2 + BE2 In Right ∆ACE …(2) Applying Pythagoras Theorem, AC2 = AE2 + CE2 …(3) …(3) By Pythagoras theorem in Δ AED AD2 = AE2 + ED2 AE2 = AD2 – ED2 By Pythagoras theorem in Δ AED AD2 = AE2 + ED2 AE2 = AD2 – ED2 AB2 + AC2 = 2AE2 + BE2 + CE2 AB2 + AC2 = 2(AD2 – ED2) + BE2 + CE2 AB2 + AC2 = 2AD2 – 2ED2 + BE2 + CE2 Now, writing BE = BD – ED CE = CD + ED AB2 + AC2 = 2AD2 – 2ED2 + (BD – ED)2 + (CD + ED)2 AB2 + AC2 = 2AD2 – 2ED2 + BD2 + ED2 – 2BD ED + CD2 + ED2 + 2 CD ED AB2 + AC2 = 2AD2 – 2ED2 + ED2 + ED2 + BD2 + CD2 + 2 CD ED – 2BD ED AB2 + AC2 = 2AD2 + BD2 + CD2 + 2 ED (CD – BD) From (1), CD = BD AB2 + AC2 = 2AD2 + BD2 + BD2 + 2 ED (BD – BD) AB2 + AC2 = 2AD2 + 2BD2 + 2 ED × 0 AB2 + AC2 = 2AD2 + 2BD2 AB2 + AC2 = 2(AD2 + 2BD2) Hence proved

CBSE Class 10 Sample Paper for 2019 Boards

Paper Summary

Question 1

Question 2 (Or 1st)

Question 2 (Or 2nd)

Question 3 (Or 1st)

Question 3 (Or 2nd)

Question 4

Question 5

Question 6

Question 7 (Or 1st)

Question 7 (Or 2nd)

Question 8 (Or 1st)

Question 8 (Or 2nd)

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16 (Or 1st)

Question 16 (Or 2nd)

Question 17 (Or 1st)

Question 17 (Or 2nd)

Question 18

Question 19 (Or 1st)

Question 19 (Or 2nd) You are here

Question 20

Question 21 (Or 1st)

Question 21 (Or 2nd)

Question 22

Question 23 (Or 1st)

Question 23 (Or 2nd)

Question 24

Question 25

Question 26

Question 27 (Or 1st)

Question 27 (Or 2nd)

Question 28 (Or 1st)

Question 28 (Or 2nd)

Question 29

Question 30

Class 10

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.