UPSC NDA Math Model Paper 7

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.
Given, BE and AD are medians of the triangle with vertices A(0, b), B(0, 0) and C(a, 0).
⇒ D is the midpoint of the side AB and E is the midpoint of the side AC
By mid point formula, we have
The coordinate of the point D=D=(

a

2

,0)
The coordinate of the point E=E=(

a

2

,

b

2

) Slope of AD=m1=

b−0

0− a 2

=

−2b

a

Slope of BE=m2=

b 2 −0

a 2 −0

=

b

a

The medians of the triangle are perpendicular to each other and the product slope of perpendicular line is -1.
⇒m1.m2=−1
⇒

−2b

a

.

b

a

=−1
⇒a2=2b2
⇒a=±√2b
The median BE and AD of a triangle with vertices A(0, b), B(0, 0) and C(a, 0) are perpendicular to each other if a=±√2b