If (a, b), (c, d) and (e, f) are the vertices of ΔABC and Δ denotes the area of ΔABC, then |acebdf111|2 is equal to ______. - Mathematics

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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 ______.

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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 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` = 4Δ2.

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