Question

In rhombus $ABCD,A{B}^2+B{C}^2+C{D}^2+D{A}^2=$ _____ .

• A. $O{A}^2+O{B}^2$
• B. $O{B}^2+O{C}^2$
• C. $O{C}^2+O{B}^2$
• D. $A{C}^2+B{D}^2$

Answer 1

The correct option is D $A{C}^2+B{D}^2$

We know that the diagonals of rhombus bisect perpendicularly.

$\therefore A{B}^2=O{A}^2+O{B}^2$

$\Rightarrow A{B}^2=\frac{A{C}^2}{4}+\frac{B{D}^2}{4}$

Similarly,

$B{C}^2=C{D}^2=D{A}^2=\frac{A{C}^2}{4}+\frac{B{D}^2}{4}$

$\therefore A{B}^2+B{C}^2+C{D}^2+D{A}^2=A{C}^2+B{D}^2$