Question

If ABCD is a parallelogram, then prove that
𝑎𝑟 (Δ𝐴𝐵𝐷) = 𝑎𝑟 (Δ𝐵𝐶𝐷) = 𝑎𝑟 (Δ𝐴𝐵𝐶) = 𝑎𝑟 (Δ𝐴𝐶𝐷) = `1/2` 𝑎𝑟 (||^𝑔𝑚 𝐴𝐵𝐶𝐷) .

Answer 1

Given: ABCDis a parallelogram
To prove : area  (Δ𝐴𝐵𝐷) = 𝑎𝑟 (ΔA𝐵𝐶) = are  (Δ ACD)

 = `1/2` 𝑎𝑟 (||𝑔𝑚 𝐴𝐵𝐶𝐷)

Proof: we know that diagonals of a parallelogram divides it into two equilaterals.
Since, AC is the diagonal.

Then, 𝑎𝑟 (Δ𝐴𝐵𝐶) =   (Δ ACD) = `1/2` 𝑎𝑟 (||^𝑔𝑚 𝐴𝐵𝐶𝐷)............ (1)

Since, BD is the diagonal

Then, 𝑎𝑟 (Δ𝐴𝐵𝐶) = 𝑎𝑟 (Δ𝐵𝐶𝐷)  = `1/2` 𝑎𝑟 (||^𝑔𝑚 𝐴𝐵𝐶𝐷)............ (2)

Compare equation (1) and (2)

∴ 𝑎𝑟 (Δ𝐴𝐵𝐶) =  𝑎𝑟 (Δ𝐴𝐶𝐷)

 = 𝑎𝑟 (Δ𝐴𝐵𝐷) =  𝑎𝑟 (Δ𝐵𝐶𝐷) =   `1/2` 𝑎𝑟 (||^𝑔𝑚 𝐴𝐵𝐶𝐷)