Determining the Base and Altitude of a 30-60-90 Triangle with a Given Hypotenuse

When dealing with a 30-60-90 triangle, the sides are in a specific and predictable ratio. Understanding these ratios can help quickly determine the lengths of the sides given the hypotenuse. This article will walk you through the process of finding the base and altitude of a 30-60-90 triangle with a hypotenuse of 16 cm.

Specifics of a 30-60-90 Triangle

In a 30-60-90 triangle:

The side opposite the 30-degree angle is denoted by Side A or simply denoted by x. The side opposite the 60-degree angle is denoted by Side B and is represented by x√3. The hypotenuse (opposite the 90-degree angle) is denoted by Side C and is twice the length of Side A, i.e., 2x.

Given Hypotenuse of 16 cm

Given that the hypotenuse is 16 cm, we can use the relationship 2x 16 to find the value of x.

Step 1: Solving for x

To solve for x, we simply divide both sides of the equation by 2:

$$ x frac{16}{2} 8 , text{cm} $$

Step 2: Finding the Lengths of Other Sides

Now that we have x 8 cm, we can determine the lengths of the other two sides:

Base (Side A) Base x 8 cm Altitude (Side B) Altitude x√3 8√3 cm ≈ 13.86 cm

Summary of Findings

In summary:

Base (opposite the 30-degree angle) 8 cm Altitude (opposite the 60-degree angle) 8√3 cm ≈ 13.86 cm

Using Trigonometric Ratios

For a more detailed understanding, we can also use the sine of the angles to find the side lengths:

Sin(30°) opposite / hypotenuse x / 16 1/2, thus x 8 cm Sin(60°) opposite / hypotenuse x√3 / 16 √3/2, thus x√3 8√3 cm

Knowing these direct relationships can simplify the process of finding side lengths in 30-60-90 triangles, making trigonometry accessible and practical.

Conclusion

In conclusion, by understanding the specific ratios of a 30-60-90 triangle and applying simple algebra, we can determine the base and altitude of a triangle given the hypotenuse length. This knowledge is not only fundamental in geometry but also has practical applications in various fields such as engineering and architecture.

Keywords

30-60-90 triangle, hypotenuse, base, altitude, trigonometry

References

[1] Math Planet