Trigonometry & Inverse Trigonometry
Properties of Triangles
Grade 11
Question:
<p>If <span class="math">\(\frac{1}{a+c} + \frac{1}{b+c} = \frac{1}{a+b+c}\)</span>, then <span class="math">\(\angle C\)</span> is</p>
<p>(a) <span class="math">\(\angle C = 75°\)</span></p>
<p>(b) <span class="math">\(\angle A = 75°\)</span></p>
<p>(c) <span class="math">\(\angle A = 60°\)</span></p>
<p>(d) <span class="math">\(\angle C = 60°\)</span></p>
Step-by-Step Solution
Key Concept: Manipulate the given algebraic condition to relate it to the cosine rule and determine the angle.
<p>Simplifying the given equation: <span class="math">\(\frac{1}{a+c} + \frac{1}{b+c} = \frac{1}{a+b+c}\)</span></p><p>Cross-multiplying and simplifying yields <span class="math">\((a+b+c)(a+b+2c) = (a+c)(b+c)\)</span>, which after algebraic manipulation gives <span class="math">\(\cos C = \frac{1}{2}\)</span>, hence <span class="math">\(C = 60°\)</span>.</p><p>∴ Answer is (d) <span class="math">\(\angle C = 60°\)</span>.</p>
Correct Answer: D