Straight Lines
Region defined by inequalities
Grade 11
Question:
<p>All the points lying inside the triangle formed by the points \((1, 3)\), \((5, 6)\), and \((-1, 2)\) satisfy:</p>
<p>(a) \(3x + 2y \geq 0\)</p>
<p>(b) \(2x + y + 1 \geq 0\)</p>
<p>(c) \(-2x + 11 \geq 0\)</p>
<p>(d) \(2x + 3y - 12 \geq 0\)</p>
Step-by-Step Solution
Key Concept: A point is inside a triangle if it satisfies all three linear inequalities corresponding to the three sides, where the inequality direction is chosen so the opposite vertex is in the satisfied region.
<p>First, find the equations of the three sides of the triangle with vertices \((1, 3)\), \((5, 6)\), and \((-1, 2)\). For each side, determine the inequality that represents the half-plane containing the opposite vertex. The intersection of these three inequalities defines the interior of the triangle. Check each option by testing all three vertices to see which inequalities are satisfied by all interior points.</p>
Correct Answer: b, c