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Question
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
2Δ2
4Δ2
2Δ
4Δ
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 4Δ2.
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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