Trigonometry & Inverse Trigonometry
Properties of Triangles
Grade 11
Question:
<p>If <span class="math">\(1 - \frac{\cos^2 A}{a^2} - 1 - \frac{\cos^2 B}{b^2} = \frac{2}{r_2} - \frac{2}{r_3}\)</span>, then the triangle is</p>
<p>(a) right angled</p>
<p>(b) equilateral</p>
<p>(c) isosceles</p>
<p>(d) None</p>
Step-by-Step Solution
Key Concept: Use exradii formulas and simplify the trigonometric condition to identify the triangle type.
<p>Simplifying the left side and using the relations for exradii <span class="math">\(r_2 = \frac{\Delta}{s-b}\)</span> and <span class="math">\(r_3 = \frac{\Delta}{s-c}\)</span>, the condition implies that two sides are equal.</p><p>∴ Answer is (c) isosceles.</p>
Correct Answer: C