Straight Lines
Coordinate Geometry and Vectors
Grade 11
Question:
<p>Let \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\) be the vertices of triangle ABC. If angle C is obtuse, then the quantity \((x_3 - x_1)(x_3 - x_2) + (y_3 - y_1)(y_3 - y_2)\) is negative.</p>
<p>(A) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1</p>
<p>(B) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1</p>
<p>(C) Statement-1 is true, Statement-2 is false</p>
<p>(D) Statement-1 is false, Statement-2 is true</p>
Step-by-Step Solution
Key Concept: The given expression is the dot product of vectors CA and CB; if angle C is obtuse, this dot product is negative.
<p><strong>Analysis:</strong> The expression \((x_3 - x_1)(x_3 - x_2) + (y_3 - y_1)(y_3 - y_2) = \vec{CA} \cdot \vec{CB}\). For angle C to be obtuse, \(\vec{CA} \cdot \vec{CB} < 0\). The dot product equals the magnitude product times cosine of the included angle, and \(\cos(\text{obtuse angle}) < 0\). Statement-2 provides geometric insight but is not the direct cause—the algebraic dot product property is.</p>
Correct Answer: A