Straight Lines
Straight Line
Allen Star Batch
Grade 11
Question:
All the points lying inside the triangle formed by the points $(1, 3), (5, 6),$ and $(-1, 2)$ satisfy:
$3x + 2y \geq 0$
$2x + y + 1 \geq 0$
$-2x + 11 \geq 0$
$2x + 3y - 12 \geq 0$
Step-by-Step Solution
Key Concept: Multiple lines intersecting at a point form angular regions whose geometric properties are analyzed using distance and coordinate relationships.
The lines intersect at origin $(0,0)$. Line $3x + 2y = 0$ passes through the origin, and the other line $2x + y + 1 = 0$ has $x$-intercept at $-\frac{1}{2}$. The vertex $(-1, 2)$ and point $(1, 3)$ are marked. The perpendicular distance from $(-1,2)$ to line $3x + 2y = 0$ determines the triangle configuration, with $x = \frac{11}{2}$ marking a key position.
Correct Answer: 1,2,3