UPSC CDS Math Triangles Questions

Consider
AB = 10,AC = 15,BC = 18
∴ BD = 9 and A'C '= 14,A'B' = 9,B'C' = 6
∴ B'D'
From Apollonius' theorem:
AB2+AC2=2(AD2+BD2)
⇒102+152=2(AD2+92)
⇒AD2=

325

2

−81=

163

2

=81.5
⇒AD=√81.5
⇒92+142=2(A′D′2+32)
⇒A′D′2=

277

2

=9
=

259

2

=129.5
⇒A′D′=√129.5
∴ AD < A'D'
Hence, statement 1 is incorrect
Remember: In a triangle, three times the sum of the squares of the sides is equal to four times the sum of the squares of the medians.
∴3(AB2+BC2+CA2)
=4(AD2+BE2+CF2)

⇒

AB2+BC2+CA2

AD2+BE2+CF2

=

4

3

.....(1)
Also,
3(A′B′2+B′C′2+C′A′2)
=4(A′D′2+B′E′2+C′F′2)
⇒

A′B′2+B′C′2+C′A′2

A′D′2+B′E′2+C′F′2

=

4

3

....(2)
From (1) and (2), we have:

AB2+BC2+CA2

AD2+BE2+CF2

=

A′B′2+B′C′2+C′A′2

A′D′2+B′E′2+C′F′2

Hence statement 2 is correct