Straight Lines
Coordinate Geometry Basics
Grade 11
Question:
<p>If the points $(0, 0)$, $(2, 2\sqrt{3})$ and $(a, b)$ are the vertices of an equilateral triangle, then $(a, b)$ is</p>
<p>(a) $(0, -4)$</p>
<p>(b) $(0, 4)$</p>
<p>(c) $(4, 0)$</p>
<p>(d) $(-4, 0)$</p>
Step-by-Step Solution
Key Concept: In an equilateral triangle, all three sides have equal length. Use the distance formula to find the required coordinates.
<p><strong>Solution:</strong> Let O$(0, 0)$, A$(2, 2\sqrt{3})$ and B$(a, b)$ be the vertices of an equilateral triangle.</p><p>For an equilateral triangle, all three sides must be equal.</p><p>First, find the distance OA:</p><p>$$OA = \sqrt{(2-0)^2 + (2\sqrt{3}-0)^2} = \sqrt{4 + 12} = \sqrt{16} = 4$$</p><p>For the triangle to be equilateral, we need $OB = AB = 4$.</p><p>Testing option (c): $B(4, 0)$</p><p>$$OB = \sqrt{4^2 + 0^2} = 4$$ ✓</p><p>$$AB = \sqrt{(4-2)^2 + (0-2\sqrt{3})^2} = \sqrt{4 + 12} = 4$$ ✓</p><p><strong>Answer: (c)</strong></p>
Correct Answer: C