Question

Show that the area of the rhombus is half the product of it lengths of its diagonals.

Answer 1

Diagonals of a rhombus are perpendicular to each other

Area of rhombus ABCD is split into area of triangle ABD

and area of triangle BCD

area of triangle $=\frac{1}{2}\times base\times height$

area of $△ABD=\frac{1}{2}\times BD\times OA$

area of $△CBD=\frac{1}{2}\times BD\times OC$

area of rhombus $=\frac{1}{2}\times BD\times OA+\frac{1}{2}\times BD\times OC$

$=\frac{1}{2}\times BD[AD+OC]=\frac{1}{2}\times BD\times AC$

$=$ half the product of length of diagonals