Question

The median of side AB of an equilateral triangle ABC is CD. The value of \( CD^2 \) will be:

• A. \( \frac{1}{2} AB^2 \)
• B. \( \frac{3}{4} AB^2 \)
• C. \( AB^2 \)
• D. \( \frac{3}{2} AB^2 \)

Hint:

In an equilateral triangle, the median is \( \frac{\sqrt{3}}{2} \times \text{side length} \), and the square of the median is \( \frac{3}{4} \times \text{side length}^2 \).

Answer 1

The Correct Option is B

Step 1: Formula for the median in an equilateral triangle.
In an equilateral triangle, the median from any vertex divides the opposite side into two equal halves and the median is also the altitude. For an equilateral triangle with side length \( a \), the median is given by: \[ \text{Median} = \frac{\sqrt{3}}{2} a \]
Step 2: Use the relationship for \( CD^2 \).
Since \( CD \) is the median in an equilateral triangle, \( CD^2 \) is given by the formula: \[ CD^2 = \frac{3}{4} AB^2 \] Thus, the value of \( CD^2 \) is \( \frac{3}{4} AB^2 \). The correct answer is (B).