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Let P (2,3), Q (x,5) and R (-1,2) be the vertices of a triangle PQR , where x > 0 . If the maximum area of a parallelogram inscribed in the triangle PQR and sharing the vertex P is 4 square units, then the value of x is ______ .

Correct answer

24

Step-by-step solution

The maximum area of a parallelogram inscribed in a triangle and sharing one of its vertices is exactly half the area of the triangle. Given that the maximum area of the inscribed parallelogram is 4 sq. units, the area of the triangle PQR must be 2 4 = 8 sq. units. The area of PQR with vertices (2,3), (x,5) and (-1,2) is given by: Area = 1 2 |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)| 8 = 1 2 |2(5 - 2) + x(2 - 3) + (-1)(3 - 5)| 16 = |2(3) + x(-1) + (-1)(-2)| 16 = |6 - x + 2| 16 = |8 - x| This gives two possible equati

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