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Question
Determine whether the point (−3, 2) lies inside or outside the triangle whose sides are given by the equations x + y − 4 = 0, 3x − 7y + 8 = 0, 4x − y − 31 = 0 .
Solution
Let ABC be the triangle of sides AB, BC and CA, whose equations x + y − 4 = 0, 3x − 7y + 8 = 0 and 4x − y − 31 = 0, respectively.
On solving them, we get \[A\left( 7, - 3 \right)\] B (2, 2) and C (9, 5) as the coordinates of the vertices.
Let P (−3, 2) be the given point.
The given point P (−3, 2) will lie inside the triangle ABC, if
(i) A and P lies on the same side of BC
(ii) B and P lies on the same side of AC
(iii) C and P lies on the same side of AB
Thus, if A and P lie on the same side of BC, then
\[\left( 21 + 21 + 8 \right)\left( - 9 - 14 + 8 \right) > 0\]
\[ \Rightarrow 50 \times \left( - 15 \right) > 0\]
\[ \Rightarrow - 750 > 0, \text { which is false }\]
Therefore, the point (−3, 2) lies outside triangle ABC.
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