Straight Lines
Reflection across a line
Grade 11
Question:
<p>Let B' be the image of point B with respect to side AC of \(\triangle ABC\). Under the same conditions, the length BB' is equal to:</p>
<p>(a) \(\frac{\sqrt{3}}{4}\)</p>
<p>(b) \(\frac{\sqrt{2}}{4}\)</p>
<p>(c) \(\frac{1}{2}\)</p>
<p>(d) \(\frac{\sqrt{3}}{2}\)</p>
Step-by-Step Solution
Key Concept: The distance between a point and its reflection across a line is twice the perpendicular distance from the point to that line.
<p><strong>Solution:</strong> B' is the reflection of B across the line AC. The distance BB' equals twice the perpendicular distance from B to AC. This perpendicular distance is the altitude from B to AC. From the triangle's properties determined in Problem 1, with AC = 1 and the angles satisfying the AP condition, the altitude from B to AC is \(\frac{\sqrt{3}}{8}\). Therefore, \(BB' = 2 \times \frac{\sqrt{3}}{8} = \frac{\sqrt{3}}{4}\).</p>
Correct Answer: a