Straight Lines
Distance formula and triangle classification
Grade 11
Question:
<p>The vertices of a triangle are <em>A</em>(4, 0), <em>B</em>(−1, −1), <em>C</em>(3, 5). The triangle is</p>
<p>Right angled and isosceles</p>
<p>Equilateral</p>
<p>Obtuse angled</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: Calculate slopes of all three sides to determine if any two are perpendicular (product of slopes = -1) or parallel (equal slopes), then verify if the triangle is right-angled, isosceles, or equilateral by checking side lengths.
<p><strong>Step 1:</strong> Find slopes of all three sides.</p><p>Slope of AB = (-1-0)/(-1-4) = -1/-5 = 1/5</p><p>Slope of BC = (5-(-1))/(3-(-1)) = 6/4 = 3/2</p><p>Slope of CA = (0-5)/(4-3) = -5/1 = -5</p><p><strong>Step 2:</strong> Check for perpendicularity by testing if any two slopes multiply to -1.</p><p>AB × CA = (1/5) × (-5) = -1 ✓</p><p>Therefore, sides AB and CA are perpendicular.</p><p><strong>Step 3:</strong> Since two sides are perpendicular, the triangle has a right angle at vertex A.</p><p>∴ Answer: A (Right-angled triangle)</p>
Correct Answer: A