Question:
<p>In a <span class="math-tex">\(\triangle A B C\)</span>, suppose <span class="math-tex">\(y=x\)</span> is the equation of the bisector of the angle B and the equation of the side <span class="math-tex">\(A C\)</span> is <span class="math-tex">\(2 x-y=2\)</span>. If <span class="math-tex">\(2 A B=B C\)</span> and the point <span class="math-tex">\(A\)</span> and <span class="math-tex">\(B\)</span> are respectively <span class="math-tex">\((4,6)\)</span> and <span class="math-tex">\((\alpha, \beta)\)</span>, then <span class="math-tex">\(\alpha+2 \beta\)</span> is equal to</p>
<p style="display:inline">42</p>
<p style="display:inline">39</p>
<p style="display:inline">45</p>
<p style="display:inline">48</p>
Step-by-Step Solution
Key Concept: Combine the Internal Angle Bisector Theorem to find the intersection point on side AC with the property that the reflection of vertex A across the angle bisector of B lies on the line BC.
<p><img src="https://media-mycbseguide.s3.amazonaws.com/images/question_images/1775188296-qzuh8k.jpg" style="height:159px; width:250px" /><br />
Point D is the bisector of AC.<br />
<span class="math-tex">$\therefore {AD}: {DC}=1: 2$</span><br />
<span class="math-tex">$\Rightarrow \frac{4-\beta}{6-\alpha}=\frac{10}{8}$</span><br />
<span class="math-tex">$\alpha=\beta$</span><br />
<span class="math-tex">$(\because y=x)$</span><br />
<span class="math-tex">$\Rightarrow \alpha=14 \text { and } \beta=14$</span><br />
<span class="math-tex">$\text { Therefore } \alpha+2 \beta=42 .$</span></p>
Correct Answer: A