Understanding the Side Length of an Equilateral Triangle with a Given Altitude

In geometry, an equilateral triangle is a special type of triangle where all three sides are equal, and all three angles are 60 degrees. When the altitude of such a triangle is known, it becomes possible to determine the side length. This article will explain the relationship between the side length and the altitude, providing a clear and detailed method for solving such problems.

Relationship Between the Altitude and the Side Length

The altitude of an equilateral triangle, denoted as h, can be related to the side length, s, through a specific formula. The altitude of an equilateral triangle with side length s is given by:

Formula

h (frac{sqrt{3}}{2}) s

This formula is derived from the properties of equilateral triangles and can be used to find either the side length or the altitude, depending on the given information. For example, if the altitude is known, the side length can be calculated by rearranging the formula.

Given Altitude and Determining the Side Length

Consider the situation where the altitude of an equilateral triangle is given as 3√3. To find the side length s, we can set up the following equation based on the formula mentioned above:

Equation Setup

3√3 (frac{sqrt{3}}{2}) s

First, we multiply both sides by 2 to simplify:

6√3 √3 s

Next, we divide both sides by √3 to isolate s:

s 6

Thus, the side length of the equilateral triangle is 6.

Explaining the 30/60/90 Triangle Relationship

An important aspect to note is the relationship between the altitude and the 30/60/90 triangle formed within the equilateral triangle. When the altitude is drawn in an equilateral triangle, it creates two 30/60/90 triangles. In a 30/60/90 triangle, the sides are in the ratio 1:√3:2. This means that if one of the sides is 1 unit, the altitude (which corresponds to (frac{√3}{2}) in the ratio) is given as:

Calculation of the Altitude in a 30/60/90 Triangle

1 x (frac{√3}{2}) (frac{√3}{2})

This confirms that for an altitude of (3√3), the side length s can be calculated as follows:

s (frac{3√3}{frac{√3}{2}})

Rearranging this formula, we get:

s 3 x 2√3/√3

Therefore, the side length is simplified to:

s 6

Additional Information and Considerations

It's worth mentioning that the altitude of an equilateral triangle can be used to find the side length through the relationship discussed. On the other hand, if the side length is known, the altitude can be determined using the same formula:

Formula Reversal

h (frac{sqrt{3}}{2}) s

Understanding these relationships and formulas is essential for solving problems related to equilateral triangles efficiently and accurately.

Conclusion

In conclusion, the side length of an equilateral triangle can be determined from its altitude using the formula h (frac{√3}{2}) s. By applying this formula and understanding the properties of 30/60/90 triangles, we can solve for the side length effectively. This knowledge is valuable not only in geometry but also in mathematics and various practical applications.