Geometry for beginners - 30 - 60 How to identify and use the "special triangle" on the right

Welcome to another topic in the Geometry for Beginners series. I am convinced that at this point of your research I realize that most of what I learned in geometry is related to finding the missing side and angle measurements. Topics such as area and volume also rely on using the required formula values. Therefore, we constantly seek short-term reductions to help find the missing value. It is important to understand that "special triangle" gives us some of the most frequently used shortcuts. The 30-60 right triangle is one of those 'special triangles'.

We have three sides of a 30-60 right triangle: (1) it is "special", (2) it makes it possible to find missing edges and angles, and (3) Find missing values.

Note: As is usually the case with geometry articles, if I can not display a picture, problems may arise for readers who need to see what is being discussed. To solve this problem, you need to create your own diagram. Please make sure you have paper and pencil when reading the geometry material.

First, why the 30-60 right triangle is "special" and where did it come from?

To understand the 30-60 right triangle, you need to check the previous topic, an equilateral triangle or conformal triangle. Let's start drawing all triangles with the same length on all three sides. This need not be exact, but closer is better. Increase the diagram to make it easier to display parts and labels. Equivalence means that all three sides are equal, equality is equiangular, and that all three sides are equal. Also note that the sum of the three angles of the triangle is always 180 degrees in total. This shows that each of the three equal angles needs to measure 60 degrees.

Add another line segment to the triangle. Drop the perpendicular segment from the top vertex to the opposite side. This segment is called the altitude of the triangle, its size is the height of the triangle.

One of the truly unique and special features of the triangle is that the altitude is median. This means that it is bisected on the opposite side in addition to being perpendicular to that plane. Besides, it bisects the apex angle into two smaller 30 degree angles. One vertical segment bisects the side and bisects an angle. Now, that's special!

Here I divided the equilateral triangle into two small right triangles. Watch your diagram again, move to the right and draw again the right triangle on the right side again. Now let's label three angles. Since the right angle makes a right angle, it is assumed that the angle is 90 degrees. The base angle to the right is from the original triangle, so its scale is 60 degrees. The upper angle is half of 60 or 30 degrees. Therefore, there are 30-60 right triangles. It is only half of the equivalent triangle.

Second: What does this triangle tell us?

Well, let's look back on your 30-60 right triangle. As an example, suppose that there is a length 4 on the hypotenuse (the opposite side of the right angle). The hypotenuse is always the longest side of a right triangle. To know this length, please tell me the length of the short side as it is exactly half of the original side or 2.

Note: The relationship between the short side and the hypotenuse is always the same, and it is represented as follows. a And 2a . Place these labels on triangles.

Now we have a right triangle with two known faces. How do we find the third aspect? you are right! In this example, 4 ^ 2 = 2 ^ 2 + b ^ 2 or 16 = 4 + b ^ 2 or 12 = b ^ 2. Therefore, b = sqrt 12 = sort (4 × 3) = 2 sqrt 3. Therefore, the three sides are 2, 2, 3, 4 in order from shortest to longest.

If more proof is required, the relationship between the three sides of the 30-60 right triangle is always a: a sqrt 3: 2a.

Third: How do you use this relationship?

Since we have to remember this relationship, we do not have to repeat the Pythagorean theorem over and over again. You also need to understand which side is which. a It is always a short side, 2a It's hypotenuse a sqrt 3 It is the other foot. Therefore, if you know that there are 30-60 right triangles, you need to know only one side to find out the other two sides using this relationship.

Example 1: A right triangle of 30 to 60 with an oblique side of 12. Find the other two sides.

solution: a: a sqrt 3: 2a Knowing that there is a relationship and 2a face is 12 means that the other face is 6 and 6 sqrt 3

In addition to the situation of finding missing edges, this relationship can also be used to identify whether the triangle is 30-60 right. Since the faces of 5, 2.5 and 2.5 sqrt 3 fit within the required ratio, the triangle must be right angled triangle and 30-60.

In conclusion, knowing this relationship and recognizing when it can be used will eliminate the need to use the Pythagoras theorem. This is a clear shortcut!