Trigonometry & Inverse Trigonometry
Triangle Properties
Grade 12
Question:
<p>The sides of triangle ABC satisfy the equation \(2a^2 + 4b^2 + c^2 - 4ab - 2ac = 0\). Then</p>
<p>(a) the triangle is isosceles</p>
<p>(b) the triangle is obtuse</p>
<p>(c) B = \(\cos^{-1}\frac{7}{8}\)</p>
<p>(d) A = \(\cos^{-1}\frac{1 + \tan\frac{n}{n}}{\cos\frac{n}{n}}\)</p>
Step-by-Step Solution
Key Concept: Factor the quadratic expression in sides to identify relationships between them.
<p>Rearranging: \(2a^2 + 4b^2 + c^2 - 4ab - 2ac = 0\)</p><p>\(= (a^2 - 4ab + 4b^2) + (a^2 - 2ac + c^2) = 0\)</p><p>\(= (a - 2b)^2 + (a - c)^2 = 0\)</p><p>This implies \(a = 2b\) and \(a = c\), so \(c = 2b\) and \(a = c\). Therefore the triangle is isosceles with \(a = c\).</p>
Correct Answer: A