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If (a, b), (c, d) and (e, f) are the vertices of ΔABC and Δ denotes the area of ΔABC, then `|(a, c, e),(b, d, f),(1, 1, 1)|^2` is equal to ______.

#### Options

2Δ

^{2}4Δ

^{2}2Δ

4Δ

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#### Solution

If (a, b), (c, d) and (e, f) are the vertices of ΔABC and Δ denotes the area of ΔABC, then `|(a, c, e),(b, d, f),(1, 1, 1)|^2` is equal to **4Δ ^{2}**.

**Explanation:**

If (a, b), (c, d) and (e, f) are vertices of ΔABC, then its area is

Δ = `1/2|(a, b, 1),(c, d, 1),(e, f, 1)|`

Δ = `1/2|(a, c ,e),(b, d, f),(1, 1, 1)|`

2Δ = `|(a, c, e),(b, d, f),(1, 1, 1)|`

On squaring both sides, we get

`\implies` (2Δ)^{2} = `|(a, c, e),(b, d, f),(1, 1, 1)|^2`

∴ `|(a, c, e),(b, d, f),(1, 1, 1)|^2` = 4Δ^{2}.

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