Determining the Side Length of an Equilateral Triangle Given Its Altitude

Understanding the relationship between the altitude and the side length of an equilateral triangle is a fundamental concept in geometry. In this article, we’ll explore how to determine the side length of an equilateral triangle when its altitude is given. We’ll start by examining the basic principles and then walk through the calculations using various methods.

Understanding the Geometry of an Equilateral Triangle

An equilateral triangle is a three-sided polygon with all sides of equal length and all internal angles measuring 60 degrees. The altitude of an equilateral triangle is a line segment drawn from a vertex perpendicular to the opposite side, effectively dividing the triangle into two congruent right triangles.

Formula for Altitude of an Equilateral Triangle

The altitude (h) of an equilateral triangle can be calculated using the formula:

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

where ( s ) is the side length of the equilateral triangle. It is important to note that this formula arises from the properties of the 30-60-90 right triangle created by the altitude.

Calculating Side Length Given Altitude

Given the altitude ( h 6 ) cm, we can use the above formula to determine the side length ( s ).

Method 1: Using the Altitude Formula

To solve for the side length ( s ), we rearrange the formula:

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

Substituting ( h 6 ) cm:

[ s frac{2 times 6}{sqrt{3}} frac{12}{sqrt{3}} 4sqrt{3} , text{cm} ]

Numerical approximation:

[ 4sqrt{3} approx 4 times 1.732 approx 6.928 , text{cm} ]

Method 2: Utilizing Trigonometry

We can also use the sine rule in a right triangle formed by the altitude. In an equilateral triangle, the base angles are 60 degrees. Thus, in the right triangle formed by the altitude, the sine of 60 degrees is given by:

[ sin 60^circ frac{6}{a} ]

From the sine rule, we know that:

[ sin 60^circ frac{sqrt{3}}{2} ]

Equating and solving for ( a ):

[ frac{sqrt{3}}{2} frac{6}{a} ]

[ a frac{2 times 6}{sqrt{3}} 4sqrt{3} , text{cm} ]

Numerical approximation:

[ 4sqrt{3} approx 6.928 , text{cm} ]

Method 3: Using the Pythagorean Theorem

Another approach involves using the Pythagorean theorem. The height ( h ) bisects the base into two equal segments, each of length ( frac{a}{2} ), and forms a right triangle with side ( a ) as the hypotenuse. Thus, in the right triangle:

[ a^2 left(frac{a}{2}right)^2 6^2 ]

[ a^2 frac{a^2}{4} 36 ]

Multiplying both sides by 4:

[ 4a^2 a^2 144 ]

[ 3a^2 144 ]

[ a^2 48 ]

[ a sqrt{48} 4sqrt{3} approx 6.93 , text{cm} ]

Conclusion

Through these methods, we have determined that the side length ( a ) of an equilateral triangle with an altitude of 6 cm is ( 4sqrt{3} ) cm or approximately 6.93 cm. Understanding these geometric relationships is crucial for solving problems in geometry and related fields.

Related Keywords

Equilateral triangle Altitude Side length