Question

For the following statement, state whether true (T) or false (F):

The sum of the squares on the sides of a rhombus is equal to the sum of the squares on its diagonals.

Answer 1

This statement is true.

Explanation:

 

Suppose ABCD is a rhombus having AC and BD its diagonals.
Since, the diagonals of a rhombus perpendicular bisect each other.
Hence, AOC is a right angle triangle
In right triangle AOC
By using Pythagoras theorem, we have 

`AB^2=((AC)/2)^2+((BD)/2)^2` 

[∴ Diagonals of a rhombus perpendicularly bisect each other] 

⇒ `AB^2=((AC)/2)^2+((BD)/2)^2` 

⇒ `4AB^2=AC^2+BD^2` 

⇒ `AB^2+AB^2+AB^2+AB^2=AC^2+BD^2` 

⇒ `AB^2+BC^2+CD^2+DA^2=AC^2+BD^2`         [∵𝐴𝑙𝑙 𝑠𝑖𝑑𝑒𝑠 𝑜𝑓 𝑎 𝑟ℎ𝑜𝑚𝑏𝑢𝑠 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙]