Straight Lines
Properties of Triangle
Grade 11
Question:
<p>A triangle with vertices (4, 0), (−1, −1), (3, 5) is</p>
<p>isosceles and right angled.</p>
<p>isosceles but not right angled.</p>
<p>right angled but not isosceles.</p>
<p>neither right angled nor isosceles.</p>
Step-by-Step Solution
Key Concept: Calculate the slopes of all three sides: if any two slopes are negative reciprocals, the triangle is right-angled. A triangle is isosceles if at least two sides have equal length.
<p><strong>Step 1:</strong> Find slopes of all three sides.</p><p>Let A = (4, 0), B = (−1, −1), C = (3, 5)</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 if any two slopes are negative reciprocals.</p><p>Slope of AB × Slope of CA = (1/5) × (−5) = −1 ✓</p><p>Therefore AB ⊥ CA, so the triangle has a right angle at vertex A.</p><p><strong>Step 3:</strong> Check if it's isosceles by computing side lengths.</p><p>|AB| = √[(−1−4)² + (−1−0)²] = √[25 + 1] = √26</p><p>|CA| = √[(4−3)² + (0−5)²] = √[1 + 25] = √26</p><p>Since |AB| = |CA|, the triangle is isosceles.</p><p>∴ The triangle is a <strong>right-angled isosceles triangle</strong> (Answer: A)</p>
Correct Answer: A