Question

A perpendicular is drawn from the vertex to the base of an equilateral triangle of side a. The measure of the perpendicular will be:

• A. \( \frac{\sqrt{3}}{2} a \) unit
• B. \( \frac{3}{2} a \) unit
• C. \( \frac{\sqrt{3}}{4} a \) unit
• D. \( \frac{3}{4} a \) unit

Hint:

In an equilateral triangle, the height is given by \( \frac{\sqrt{3}}{2} \times \text{side length} \).

Answer 1

The Correct Option is A

Step 1: Understanding the geometry of an equilateral triangle.
In an equilateral triangle, the height (or perpendicular) divides the triangle into two 30-60-90 right triangles. For a 30-60-90 triangle, the ratio of the sides opposite to the 30°, 60°, and 90° angles is \( 1 : \sqrt{3} : 2 \).
Step 2: Applying the geometry to the equilateral triangle.
Let the side of the equilateral triangle be \( a \). The height of the triangle is the side opposite to the 60° angle in the 30-60-90 triangle. The ratio of the sides gives: \[ \frac{\text{height}}{a/2} = \sqrt{3} \] Thus, the height is: \[ \text{height} = \frac{\sqrt{3}}{2} \times a \] Therefore, the measure of the perpendicular (height) is \( \frac{\sqrt{3}}{2} a \) units. The correct answer is (A).