Introduction to the Problem

Given that an equilateral triangle has a perpendicular height of 15 cm, we aim to find the length of its sides, correct to two decimal places. This problem involves applying geometric principles and algebraic manipulation to derive the solution.

Understanding the Geometry of an Equilateral Triangle

An equilateral triangle is a unique type of triangle where all three sides are of equal length, and consequently, all three angles are also equal at 60 degrees each. The perpendicular height, or altitude, of an equilateral triangle splits it into two 30-60-90 right-angled triangles. This relationship is crucial for solving the problem at hand.

Method 1: Using the Height-Altitude Formula

The formula for the height ( h ) of an equilateral triangle in terms of its side length ( s ) is:

[ h frac{sqrt{3}}{2}s ]

Given ( h 15 ) cm, we rearrange the formula to solve for ( s ):

[ s frac{2h}{sqrt{3}} ]

Substituting the given height:

[ s frac{2 times 15}{sqrt{3}} frac{30}{sqrt{3}} 30 times frac{sqrt{3}}{3} 10sqrt{3} ]

Using the approximate value of ( sqrt{3} approx 1.732 ):

[ s approx 10 times 1.732 17.32 text{ cm} ]

Thus, the length of each side of the equilateral triangle is approximately:

[ boxed{17.32 text{ cm}} ]

Method 2: Using the Sine Rule

The sine rule is a fundamental property of triangles, and it can also be used to solve this problem. Let ( x ) be the side length of the triangle. Given that the angle opposite the height (15 cm) is 60 degrees, and the angle opposite the side length is 90 degrees, we can set up the sine rule as follows:

[ frac{15}{sin 60^circ} frac{x}{sin 90^circ} ]

Solving for ( x ):

[ x 15 cdot frac{sin 90^circ}{sin 60^circ} ]

Since ( sin 90^circ 1 ) and ( sin 60^circ frac{sqrt{3}}{2} ):

[ x 15 cdot frac{1}{frac{sqrt{3}}{2}} 15 cdot frac{2}{sqrt{3}} frac{30}{sqrt{3}} 10sqrt{3} ]

Thus,

[ x approx 10 times 1.732 17.32 text{ cm} ]

Therefore, the length of each side of the equilateral triangle is approximately:

[ boxed{17.32 text{ cm}} ]

Alternative Approach: Using the Pythagorean Theorem

Using the Pythagorean Theorem, we can also derive the side length. Given that the height bisects the 60-degree angle into two 30-degree angles and splits the opposite side into two equal segments, we can set up the following equation:

[ x^2 left(frac{x}{2}right)^2 15^2 ]

Solving for ( x ):

[ x^2 frac{x^2}{4} 225 ]

[ x^2 - frac{x^2}{4} 225 ]

[ frac{3x^2}{4} 225 ]

[ x^2 frac{225 times 4}{3} 300 ]

[ x sqrt{300} 10sqrt{3} approx 17.32 text{ cm} ]

Therefore, the length of each side of the equilateral triangle is approximately:

[ boxed{17.32 text{ cm}} ]

Conclusion

Regardless of the method used, the length of each side of the equilateral triangle is found to be 17.32 cm (correct to two decimal places). This problem demonstrates the application of geometric principles and trigonometric relations to solve real-world problems involving shapes and measurements.