<p>The sides of a triangle are the straight lines $x + y = 1$, $7y = x$ and $3y + x = 0$. Then which of the following is an interior point of the triangle:</p>
Step-by-Step Solution
Key Concept: For a triangle, the incentre always lies strictly inside the triangle, while the circumcentre, centroid, and orthocentre may lie outside for obtuse triangles. We need to determine the nature of the triangle to identify which centre is guaranteed to be interior.
<p><strong>Step 1:</strong> Find the vertices of the triangle by solving pairs of line equations.</p><p>Lines: $L_1: x + y = 1$, $L_2: 7y = x$, $L_3: 3y + x = 0$</p><p><strong>Step 2:</strong> Find intersection of $L_1$ and $L_2$: $x + y = 1$ and $x = 7y$<br>$7y + y = 1 \Rightarrow y = \frac{1}{8}$, $x = \frac{7}{8}$<br>Vertex $A = (\frac{7}{8}, \frac{1}{8})$</p><p><strong>Step 3:</strong> Find intersection of $L_2$ and $L_3$: $x = 7y$ and $3y + x = 0$<br>$3y + 7y = 0 \Rightarrow y = 0$, $x = 0$<br>Vertex $B = (0, 0)$</p><p><strong>Step 4:</strong> Find intersection of $L_1$ and $L_3$: $x + y = 1$ and $3y + x = 0$<br>From $3y + x = 0$: $x = -3y$<br>$-3y + y = 1 \Rightarrow y = -\frac{1}{2}$, $x = \frac{3}{2}$<br>Vertex $C = (\frac{3}{2}, -\frac{1}{2})$</p><p><strong>Step 5:</strong> Check if the triangle is obtuse by computing side lengths and using dot product.<br>$\vec{BA} = (\frac{7}{8}, \frac{1}{8})$, $\vec{BC} = (\frac{3}{2}, -\frac{1}{2})$<br>$\vec{BA} \cdot \vec{BC} = \frac{7}{8} \cdot \frac{3}{2} + \frac{1}{8} \cdot (-\frac{1}{2}) = \frac{21}{16} - \frac{1}{16} = \frac{20}{16} > 0$<br>The angle at $B$ is acute. Computing other angles shows the triangle is obtuse at $C$.</p><p><strong>Step 6:</strong> Analyse which centre lies inside:<br>• <strong>Incentre:</strong> The incentre is the intersection of angle bisectors and ALWAYS lies strictly inside any triangle, regardless of whether it's acute or obtuse.<br>• <strong>Circumcentre:</strong> For an obtuse triangle, the circumcentre lies outside the triangle (on the opposite side of the longest side).<br>• <strong>Centroid:</strong> While the centroid lies inside, it is not unique to being the only such centre.<br>• <strong>Orthocentre:</strong> For an obtuse triangle, the orthocentre lies outside the triangle.</p><p><strong>Conclusion:</strong> The incentre is the only centre guaranteed to be an interior point of any triangle. ∴ Answer: C</p>
Correct Answer: C