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) :
Show Hint
In an equilateral triangle, the altitude is always \(\frac{\sqrt{3}}{2}\) times the side length.
If the altitude is \(\frac{5\sqrt{3}}{2}\), then the side must be 5.
Since the base is bisected by the origin, the coordinates are simply half of the side length on either side of the origin, giving \(\pm \frac{5}{2}\).
This simple mental calculation helps you find the correct coordinates quickly.
Step 1: Understanding the Question:
The topic of this question is Coordinate Geometry combined with the geometric properties of Equilateral Triangles.
We are given an equilateral triangle \( ABC \) where \( AD \) is a median.
The coordinates of vertex \( A \) are given as \(\left(0, \frac{5\sqrt{3}}{2}\right)\), and the point \( D \) is at 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 from a vertex to the opposite side is also the altitude (perpendicular bisector) of that side.
- The length of the altitude \( h \) of an equilateral triangle with side length \( a \) is given by the formula:
\[ h = \frac{\sqrt{3}}{2}a \]
- Distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated as:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 3: Detailed Explanation:
1. First, let's analyze the alignment of the triangle.
Since vertex \( A \) lies on the y-axis (as its x-coordinate is 0) and \( D \) is at the origin \((0, 0)\), the median \( AD \) lies along the y-axis.
Since the median is perpendicular to the base \( BC \) in an equilateral triangle, the base \( BC \) must lie along the x-axis.
Since \( D(0, 0) \) is the midpoint of \( BC \), the vertices \( B \) and \( C \) must be symmetric with respect to the origin.
2. Let the coordinates of \( B \) be \((-x, 0)\) and \( C \) be \((x, 0)\).
The length of side \( BC \) is \( 2x \). Let the side length of the equilateral triangle be \( a \). Thus, \( a = 2x \), which means \( x = \frac{a}{2} \).
3. The length of the median \( AD \) represents the height \( h \) of the triangle.
Using the coordinates of \( A\left(0, \frac{5\sqrt{3}}{2}\right)\) and \( D(0, 0)\), the height \( h \) is:
\[ h = \sqrt{(0 - 0)^2 + \left(\frac{5\sqrt{3}}{2} - 0\right)^2} = \frac{5\sqrt{3}}{2} \]
4. Equate this height to the equilateral triangle altitude formula:
\[ \frac{\sqrt{3}}{2}a = \frac{5\sqrt{3}}{2} \]
5. Solving for side length \( a \) by canceling \(\frac{\sqrt{3}}{2}\) on both sides:
\[ a = 5 \]
6. Since \( B \) and \( C \) lie on the x-axis and are symmetric about the origin, their distance from the origin is:
\[ x = \frac{a}{2} = \frac{5}{2} \]
Point \( B \) lies on the negative x-axis, so its coordinates are \(\left(-\frac{5}{2}, 0\right)\).
Point \( C \) lies on the positive x-axis, so its coordinates are \(\left(\frac{5}{2}, 0\right)\).
Step 4: Final Answer:
The coordinates of points B and C are \(\left(-\frac{5}{2}, 0\right)\) and \(\left(\frac{5}{2}, 0\right)\), which corresponds to option (B).
