The side of an equilateral triangle is 14 cm. What is the area of the triangle in square centimetres?

Introduction / Context:
This problem asks for the area of an equilateral triangle when the side length is known. It tests familiarity with the standard area formula for equilateral triangles as well as skill in substituting and simplifying expressions involving square roots.

Given Data / Assumptions:

• The triangle is equilateral with all three sides equal.
• Side length a = 14 cm.
• We need the area expressed in terms of square centimetres, possibly with a square root in the final answer.

Concept / Approach:
The area A of an equilateral triangle with side a is given by the formula A = (sqrt(3) / 4) * a^2. This formula comes from drawing an altitude, splitting the triangle into two right triangles, and using Pythagoras theorem. Once the formula is recalled, we simply substitute a = 14 cm and simplify.

Step-by-Step Solution:
Step 1: Recall the area formula: A = (sqrt(3) / 4) * a^2.Step 2: Substitute a = 14 cm: A = (sqrt(3) / 4) * 14^2.Step 3: Compute 14^2 = 196.Step 4: So A = (sqrt(3) / 4) * 196.Step 5: Simplify the fraction: 196 / 4 = 49.Step 6: Therefore A = 49 * sqrt(3) square centimetres, written as 49√3.

Verification / Alternative check:
We can also compute the height of the equilateral triangle and use the standard triangle area formula A = (1 / 2) * base * height.The height h equals (sqrt(3) / 2) * a, so for a = 14 cm, h = (sqrt(3) / 2) * 14 = 7 * sqrt(3).Then A = (1 / 2) * base * height = (1 / 2) * 14 * 7 * sqrt(3) = 7 * 7 * sqrt(3) = 49√3.This matches the previous result, confirming the correctness of the answer.

Why Other Options Are Wrong:
Options like 64√3 or 64√2 involve incorrect use of the formula or incorrect squaring of the side length.An option such as 49√2 would result from using a wrong height or misapplying Pythagoras theorem.Only 49√3 matches the exact computation based on the standard area formula for an equilateral triangle of side 14 cm.

Common Pitfalls:
A common mistake is to use the formula A = a^2 * sqrt(3) instead of A = (sqrt(3) / 4) * a^2, which overestimates the area.Another error is squaring 14 incorrectly or forgetting to divide by 4 after multiplying by a^2.Some learners may also mix up inradius or circumradius formulas with the area formula, which leads to incorrect options.

Final Answer:
The area of the equilateral triangle is 49√3 square centimetres.