If triangle ABC is an equilateral triangle with each side 16 cm long, what is the length of its altitude?

Introduction / Context:This problem is about an equilateral triangle and its altitude. In an equilateral triangle, all sides are equal and all angles are 60 degrees. The altitude has a special relationship with the side length because it also acts as a median and an angle bisector. Knowing the formula for the altitude of an equilateral triangle is very useful for geometry and trigonometry questions in aptitude tests.

Given Data / Assumptions:• Triangle ABC is equilateral.• Each side of triangle ABC = 16 cm.• We are required to find the length of an altitude of this triangle.

Concept / Approach:In an equilateral triangle with side a, the altitude h can be found using Pythagoras theorem. When the altitude is drawn from a vertex to the opposite side, it splits the triangle into two congruent right angled triangles. Each has hypotenuse a and base a/2. Therefore, h^2 + (a/2)^2 = a^2. Solving gives h = (sqrt(3) / 2) * a. We can directly apply h = (sqrt(3) / 2) * a to compute the altitude length for side 16 cm.

Step-by-Step Solution:Step 1: Use the altitude formula for an equilateral triangle.h = (sqrt(3) / 2) * a.Step 2: Substitute the given side length a = 16 cm.h = (sqrt(3) / 2) * 16.Step 3: Simplify the expression.16 / 2 = 8.So h = 8 * sqrt(3) cm.

Verification / Alternative check:We can verify by using Pythagoras theorem directly. After drawing the altitude, we have two right triangles with hypotenuse 16 cm and one leg 8 cm (half of the base). Then h^2 = 16^2 − 8^2 = 256 − 64 = 192. So h = sqrt(192). Factoring 192 as 64 * 3, we get h = sqrt(64 * 3) = 8 * sqrt(3), which agrees with our formula based calculation.

Why Other Options Are Wrong:Option A: 2√3 cm would correspond to a much smaller triangle and does not match side 16 cm.Option B: 4√3 cm is exactly half the correct altitude and would match a triangle with side 8 cm.Option D: 5√3 cm does not result from any correct computation based on the given side length.

Common Pitfalls:Candidates may confuse the formula for altitude with that of the median in other triangles or may misremember the coefficient, writing a/2 instead of (sqrt(3) / 2) * a. Errors also occur when simplifying the square root of 192. Remembering that in an equilateral triangle the altitude is greater than half the side and using h = (sqrt(3) / 2) * a helps prevent such mistakes.

Final Answer:The length of the altitude of the equilateral triangle is 8√3 cm.