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Question
If ABCD is a parallelogram, then prove that
๐๐ (Δ๐ด๐ต๐ท) = ๐๐ (Δ๐ต๐ถ๐ท) = ๐๐ (Δ๐ด๐ต๐ถ) = ๐๐ (Δ๐ด๐ถ๐ท) = `1/2` ๐๐ (||๐๐ ๐ด๐ต๐ถ๐ท) .
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Solution
Given: ABCDis a parallelogram
To prove : area (Δ๐ด๐ต๐ท) = ๐๐ (ΔA๐ต๐ถ) = are (Δ ACD)
= `1/2` ๐๐ (||๐๐ ๐ด๐ต๐ถ๐ท)
Proof: we know that diagonals of a parallelogram divides it into two equilaterals.
Since, AC is the diagonal.
Then, ๐๐ (Δ๐ด๐ต๐ถ) = (Δ ACD) = `1/2` ๐๐ (||๐๐ ๐ด๐ต๐ถ๐ท)............ (1)
Since, BD is the diagonal
Then, ๐๐ (Δ๐ด๐ต๐ถ) = ๐๐ (Δ๐ต๐ถ๐ท) = `1/2` ๐๐ (||๐๐ ๐ด๐ต๐ถ๐ท)............ (2)
Compare equation (1) and (2)
∴ ๐๐ (Δ๐ด๐ต๐ถ) = ๐๐ (Δ๐ด๐ถ๐ท)
= ๐๐ (Δ๐ด๐ต๐ท) = ๐๐ (Δ๐ต๐ถ๐ท) = `1/2` ๐๐ (||๐๐ ๐ด๐ต๐ถ๐ท)
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