Question

A B C D is quadrilateral whose diagonal AC divides it into two equal parts .show that AC bisects BD

Answer 1

Solution:

Given : A quadrilateral AB CD is such that Diagonal AC divides it into two equal parts.

To prove : Diagonal AC bisects B D.

Construction : Draw perpendicular bisector of Diagonal AC from vertex B as well as D, such that it intersects Diagonal A C at M and N respectively.  

Proof : Area of a triangle = $\frac{1}{2}\times {\text{Base}} \times {\text{Height}}$

Area (ΔABC)=  $\frac{1}{2} \times {BM} \times {AC}$

Area (ΔADC)=  $\frac{1}{2} \times {D N} \times {AC}$

As ,  Area (ΔABC)=  Area (ΔADC)  →→→(Given)

which gives, BM= D N

It means point M and N are Collinear, Showing

Diagonal A C divides or bisects B D in two equal parts.