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