Key Properties of 30-60-90 Triangles and the Pythagorean Theorem

A 30-60-90 triangle is a kind of right triangle. It has three angles:

• One angle is 30 degrees.
• Another angle is 60 degrees.
• The last angle is 90 degrees.

Because of its special angles, this triangle is very useful in geometry.

Ratio of Sides

The most important feature of a 30-60-90 triangle is the ratio of its sides. Here’s how it breaks down:

• The side across from the 30-degree angle (the shortest side) is called $x$.
• The side across from the 60-degree angle (the longer side) is $x\sqrt{3}$.
• The hypotenuse (across from the 90-degree angle) is $2x$.

So, the sides of a 30-60-90 triangle always have this ratio:

$$1 : \sqrt{3} : 2$$

This means if you know one side, you can easily find the others.

Relationship to the Pythagorean Theorem

The Pythagorean Theorem helps us understand right triangles. It says that, in any right triangle, the sum of the squares of the two shorter sides (the legs) equals the square of the longest side (the hypotenuse). We can write this as:

$$a^2 + b^2 = c^2$$

Where:

• $a$ and $b$ are the lengths of the two shorter sides.
• $c$ is the length of the hypotenuse.

For a 30-60-90 triangle, we can use:

• $a = x$ (the side across from the 30-degree angle),
• $b = x\sqrt{3}$ (the side across from the 60-degree angle),
• $c = 2x$ (the hypotenuse).

Plugging these into the Pythagorean theorem gives us:

$$x^2 + (x\sqrt{3})^2 = (2x)^2$$

If we simplify this, we get:

$$x^2 + 3x^2 = 4x^2$$

This simplifies to:

$$4x^2 = 4x^2$$

This shows that the 30-60-90 triangle follows the Pythagorean theorem.

Applications and Examples

Knowing about 30-60-90 triangles helps us in real-life situations. Here are a few examples:

• Construction: Builders use these triangles to make sure they have right angles. They can easily find the lengths of the sides if they know just one side.

• Trigonometry: The sine, cosine, and tangent values for the angles can be found easily:

  • For 30 degrees:
    • $\sin(30^\circ) = \frac{1}{2}$
    • $\cos(30^\circ) = \frac{\sqrt{3}}{2}$
    • $\tan(30^\circ) = \frac{1}{\sqrt{3}}$
  • For 60 degrees:
    • $\sin(60^\circ) = \frac{\sqrt{3}}{2}$
    • $\cos(60^\circ) = \frac{1}{2}$
    • $\tan(60^\circ) = \sqrt{3}$

Here's a simple example: If the shortest side (across from 30 degrees) is 3 units, then:

• The side across from 60 degrees is $3\sqrt{3}$, which is about 5.20 units.
• The hypotenuse is $2 \times 3 = 6$ units.

Conclusion

In summary, the 30-60-90 triangle has clear properties that relate closely to the Pythagorean theorem. This helps us do calculations in geometry more easily. Knowing how to work with these triangles is important for solving many math problems, especially for students in Grade 9 in the United States.