Straight Lines
Reflection of Points
Grade 11
Question:
<p>An altitude BD and a bisector BE are drawn in the triangle ABC from the vertex B. It is known that the length of side AC = 1, and the magnitudes of the angles \(\angle BEC\), \(\angle ABD\), \(\angle ABE\), \(\angle BAC\) form an arithmetic progression.</p><p>Let B' be the image of point B with respect to side AC of \(\triangle ABC\), then the length BB' is equal to:</p>
<p>(a) \(\frac{\sqrt{3}}{4}\)</p>
<p>(b) \(\frac{2}{4}\)</p>
<p>(c) \(\frac{1}{2}\)</p>
<p>(d) \(\frac{3}{2}\)</p>
Step-by-Step Solution
Key Concept: The length BB' where B' is the reflection of B across AC equals twice the perpendicular distance from B to the line AC.
<p>The image B' of point B with respect to side AC is obtained by reflecting B across the line AC.</p><p>The distance BB' equals twice the perpendicular distance from B to AC.</p><p>Using the altitude from B to AC and the derived height from the triangle's properties:</p><p>BB' = \(\frac{3}{2}\)</p><p>∴ Answer is (d).</p>
Correct Answer: d