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) :
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For any equilateral triangle with base on the x-axis and vertex on the positive y-axis:
The x-coordinates of the base vertices are always $\pm \frac{h}{\sqrt{3}}$, where $h$ is the altitude.
Here, $h = \frac{5\sqrt{3}}{2}$.
So the x-coordinates are:
\[ \pm \frac{\frac{5\sqrt{3}}{2}}{\sqrt{3}} = \pm \frac{5}{2} \]
This quick shortcut directly yields the base coordinates in one line!
Step 1: Understanding the Question:
This question combines concepts from "Coordinate Geometry" and properties of "Equilateral Triangles".
We are given an equilateral triangle $ABC$.
The median $AD$ joins the vertex $A\left(0, \frac{5\sqrt{3}}{2}\right)$ to the midpoint of the base $D(0, 0)$.
We need to find the coordinates of the other two vertices, $B$ and $C$.
Step 2: Key Formula or Approach:
We use the special properties of an equilateral triangle:
• In an equilateral triangle, the median drawn to the base is also the altitude. Therefore, $AD \perp BC$.
• Since $D(0, 0)$ is the origin and $A$ lies on the y-axis ($x = 0$), the altitude $AD$ lies along the y-axis.
• Because the altitude is perpendicular to the base, the base $BC$ must lie along the x-axis ($y = 0$).
• The length of the altitude $h$ of an equilateral triangle with side length $a$ is given by:
\[ h = \frac{\sqrt{3}}{2} a \]
• Since $D$ is the midpoint of $BC$, the distance from $D$ to $B$ and $D$ to $C$ is half of the side length ($\frac{a}{2}$).
Step 3: Detailed Explanation:
• Calculate the length of the altitude $AD$ using the coordinates of $A\left(0, \frac{5\sqrt{3}}{2}\right)$ and $D(0,0)$:
\[ h = \text{Distance } AD = \frac{5\sqrt{3}}{2} - 0 = \frac{5\sqrt{3}}{2} \]
• Let the side length of the equilateral triangle be $a$. The relationship between the altitude and the side length is:
\[ h = \frac{\sqrt{3}}{2} a \]
• Substitute the value of $h$ into this formula:
\[ \frac{5\sqrt{3}}{2} = \frac{\sqrt{3}}{2} a \]
• Solve for the side length $a$:
By equating both sides, we immediately get:
\[ a = 5 \]
So the side length of the equilateral triangle is 5 units.
• Since the base $BC$ lies on the x-axis and $D(0,0)$ is the midpoint, the points $B$ and $C$ lie symmetrically on either side of the origin along the x-axis.
• The distance of $B$ and $C$ from $D(0,0)$ is half of the side length:
\[ \text{Distance} = \frac{a}{2} = \frac{5}{2} \]
• Therefore, the coordinates are:
- Point $B$ (to the left of origin): $\left(-\frac{5}{2}, 0\right)$
- Point $C$ (to the right of origin): $\left(\frac{5}{2}, 0\right)$
Step 4: Final Answer:
The coordinates of $B$ and $C$ are $\left(-\frac{5}{2}, 0\right)$ and $\left(\frac{5}{2}, 0\right)$, which matches Option (B).
