Find the length of the altitude of an equilateral triangle whose side length is 33 cm. Give the answer in centimetres, rounded to one decimal place if needed.

Introduction / Context:
This question focuses on properties of an equilateral triangle, in which all sides and all angles are equal. The altitude of an equilateral triangle is a special line segment that serves as height, median, and perpendicular bisector. Aptitude questions frequently use the standard altitude formula for equilateral triangles to test familiarity with basic geometric formulas and comfort with simple substitution and calculation.

Given Data / Assumptions:

• The triangle is equilateral, so all three sides are equal and each interior angle is 60°.
• Each side of the triangle has length 33 cm.
• We are asked to find the length of the altitude drawn from any vertex to the opposite side.
• We will use the standard formula for altitude of an equilateral triangle: h = (√3 / 2) * side.
• For a numerical answer, we approximate √3 as 1.732 and round to one decimal place for simplicity.

Concept / Approach:
In an equilateral triangle, drawing an altitude splits the triangle into two congruent right triangles. Each right triangle has hypotenuse equal to the side of the equilateral triangle and one acute angle of 60°. Using basic trigonometry or the known formula, the altitude equals (√3 / 2) times the side length. Substituting the given side length into this expression gives the desired height. This approach is standard and avoids any complicated geometry.

Step-by-Step Solution:
Let the side length be a = 33 cm.Altitude of an equilateral triangle is h = (√3 / 2) * a.Substitute a = 33: h = (√3 / 2) * 33.Compute numerically: h ≈ (1.732 / 2) * 33 ≈ 0.866 * 33 ≈ 28.578 cm.Rounded to one decimal place, h ≈ 28.6 cm.

Verification / Alternative check:
Another way is to use right triangle relations. If we drop the altitude from one vertex, the base side of length 33 cm is split into two equal halves of 16.5 cm. Each right triangle then has hypotenuse 33 cm and one leg 16.5 cm. Using Pythagoras theorem, altitude h satisfies 33^2 = 16.5^2 + h^2. So h^2 = 33^2 − 16.5^2 = 1089 − 272.25 = 816.75, and h ≈ √816.75 ≈ 28.6 cm. This matches the earlier computation, confirming the result.

Why Other Options Are Wrong:
Option 27.5 cm underestimates the height and does not satisfy the Pythagoras relation with side 33 cm. Option 26.4 cm is also too small and would produce an area that is significantly lower than the correct area of the equilateral triangle. Option 30.0 cm is larger than the altitude of a 33 cm equilateral triangle and does not satisfy h^2 = 33^2 − 16.5^2. Option 24.8 cm is even smaller and inconsistent with the basic geometric formula for an equilateral triangle altitude.

Common Pitfalls:
Many students confuse the altitude formula with the area formula or forget the factor of 1/2 in (√3 / 2) * side. Some might mistakenly compute (side * side) / 2 instead of using the correct formula. Others forget that the altitude is longer than half the side but shorter than the full side, leading them to pick values that are obviously too small or too large. Using an incorrect approximation for √3 can also introduce noticeable error.

Final Answer:
The length of the altitude of the equilateral triangle is approximately 28.6 cm.