Question

prove that area of square is equal to the half product of its diagonal.​

Answer 1

Answer:

Area of a rhombus =  12×d1×d2  

Square is a rhombus (albeit , of a special kind)

Hence, Proved.

Aliter:

Length of diagonal of square =  2–√a  

Product of diagonals =  2a2  

Half of product of diagonals =  a2  = Area of square

Hence, Proved.

Answer 2

Answer:

Step-by-step explanation: Let ABCD is a square, each side is x unit. Diagonals AC =BD =y unit.If diagonals intersect at point O. Angle AOB=90° and OA=OB= y/2.

In right angled triangle AOB

OA^2+OB^2=AB^2

(y^2)/4+(y^2)/4=x^2 or x^2=(y^2)/2…………..(1)

Area of square=(side)^2=(x)^2 , [put x^2=(y^2)/2 from eq.(1).]

Area of square =(y^2)/2=(1/2)×y×y=(1/2)AC×BD.

= Half of the product of diagonals. Proved.