Question

In the given figure, \(\Delta ABC\) is an equilateral triangle. \(AD\) is a median of the triangle joining the points \(A\left(0, \frac{5\sqrt{3}}{2}\right)\), \(D(0, 0)\). Points \(B\) and \(C\) are (in same order) :

• A. \((- 5, 0), (5, 0)\)
• B. \(\left(-\frac{5}{2}, 0\right), \left(\frac{5}{2}, 0\right)\)
• C. \((- 10, 0), (10, 0)\)
• D. \((- 5\sqrt{3}, 0), (5\sqrt{3}, 0)\)

Hint:

For any equilateral triangle symmetric about an axis, the height and half-base form a \(30^\circ - 60^\circ - 90^\circ\) right-angled triangle.
The ratio of sides in such a triangle is \(1 : \sqrt{3} : 2\).
Since the height is \(\frac{5}{2}\sqrt{3}\), the half-base is simply \(\frac{5}{2}\), which instantly gives the x-coordinates.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given an equilateral triangle \(\Delta ABC\) with its median \(AD\).
The coordinates of vertex \(A\) are \(\left(0, \frac{5\sqrt{3}}{2}\right)\) and the point \(D\) is the origin \((0, 0)\).
We need to determine the coordinates of the other two vertices, \(B\) and \(C\).

Step 2: Key Formula or Approach:
In an equilateral triangle, the median drawn to a side is also the perpendicular bisector of that side.
Since the median \(AD\) lies along the y-axis (as both \(A\) and \(D\) have an x-coordinate of \(0\)), the base \(BC\) must lie along the x-axis.
The height \(h\) of an equilateral triangle with side length \(a\) is given by:
\[ h = \frac{\sqrt{3}}{2}a \]
Since \(D(0,0)\) is the midpoint of \(BC\), the distance from \(D\) to \(B\) and from \(D\) to \(C\) is half of the side length, i.e., \(\frac{a}{2}\).

Step 3: Detailed Explanation:
1. The length of the median \(AD\) is the distance between \(A\left(0, \frac{5\sqrt{3}}{2}\right)\) and \(D(0,0)\).
\[ AD = \frac{5\sqrt{3}}{2} - 0 = \frac{5\sqrt{3}}{2} \]
2. Since \(AD\) is the altitude (height \(h\)) of the equilateral triangle, we equate it to the height formula:
\[ \frac{\sqrt{3}}{2}a = \frac{5\sqrt{3}}{2} \]
3. Solving for the side length \(a\):
\[ a = 5 \]
4. The vertices \(B\) and \(C\) lie on the x-axis symmetric to the origin \(D(0,0)\).
5. Therefore, the distance of both \(B\) and \(C\) from the origin \(D\) is half of the side length:
\[ OD = \frac{a}{2} = \frac{5}{2} \]
6. Since \(B\) lies on the negative x-axis and \(C\) lies on the positive x-axis, their coordinates are:
- \(B = \left(-\frac{5}{2}, 0\right)\)
- \(C = \left(\frac{5}{2}, 0\right)\)
7. This matches option (B).

Step 4: Final Answer:
The correct option is (B).