Q16 of 168 Page 4

In an acute-angled triangle, express a median in terms of its sides.

16.jpg

We have

In ABC, AD is median

AE⊥BC

In AEB

AB2=AE2+BE2

AB2=AD2-DE2+(BD-DE)2

AB2=AD2-DE2+BD2-2xBDxDE+DE2

AB2=AD2+BD2-2xBDxDE

AB2=AD2+BC2/4-BCxDE …………. (I) [GIVEN BC=2BD]

In AEC

AC2=AE2+EC2

AC2=AD2-DE2+ (DE+CD)2

AC2=AD2-DE2+2CDxDE

AC2=AD2+BC2/4+BCxDE ……….(II) [BC=2CD]

By adding equ. (i) and (ii) we get

AB2+AC2=2AD2+BC2/2

2AB2+2AC2=4AD2+BC2 [MULTIPLY BY 2]

4AD2=2AB2+2AC2-BC2

AD2=2AB2+2AC2-BC2

 

More from this chapter

All 168 →
14

The lengths of the diagonals of a rhombus are 24 cm and 10 cm. Find each side of the rhombus.

 15

Each side of a rhombus is 10 cm. If one of its diagonals is 16 cm find the length of the other diagonal.

 17

Calculate the height of an equilateral triangle each of whose sides measures 12 cm.

 18

In right-angled triangle ABC in which, if D is the mid-point of BC, prove that .