Geometry for beginners - how to define and use the right "special triangle" 30-60,

Welcome to another topic in the Geometry for Beginners series. At this point in your research, I am sure you understand that most of what is studied in geometry is related to the search for missing dimensions for the sides and angles of numbers. Even with topics such as area and volume, we depend on the availability of the necessary values ​​of the formulas. Thus, we are always looking for short cuts to help us find the missing values. “Special triangles” are important to understand because they give us some of the most commonly used labels. The right triangle 30-60 is one of those “special triangles”.

We will look at three aspects of the right triangle 30-60: (1) why it is “special”, (2) what it tells us to allow the search for missing sides and angles, and (3) how we actually use it to find missing values.

Note. As is usually the case in the Geometry articles, my inability to present diagrams can cause problems for the reader who needs to LOOK what is being discussed. To overcome this problem, you need to create your own charts. make sure you have paper and pencil when you read geometry materials.

First: Why is the right triangle 30-60 “special” and where does it come from?

To understand the right triangle 30-60, we need to consider the previous topic - an equilateral or equilateral triangle. First, let's start drawing a triangle with all three sides of the same length. It doesn’t have to be exact, but the closer the better. Make the diagram large enough so you can easily see the details and labels. Remember that the equivalent means that all 3 sides are equal and equivalent are also equivalent, which means that all 3 corners are equal. Also remember that the sum of all three corners of a triangle is always 180 degrees. This suggests that each of the three equal angles should measure 60 degrees.

We are going to add another line segment to our triangle. From the upper vertex, move the perpendicular line segment to the opposite side. This segment is called the height of the triangle, and its measure is the height of the triangle.

One of the unique and unique features of a one-sided triangle is that the height is also a median. This means that it halves (divides into 2 equal parts) the opposite side in addition to being perpendicular to that side. In addition, he also halves the upper angle into two smaller angles of 30 degrees. One segment, perpendicular, halves the side and halves the angle. Now, this is special!

Now we have divided our equilateral triangle into two smaller right triangles. Once again looking at the picture, move to the right and again draw the right triangle, which is on the right. Now let's denote the three corners. We know that the perpendicular forms a right angle, so we denote this angle as 90 degrees. The base angle is to the right of the original triangle, so its size is 60 degrees; and the top corner is half 60 or 30 degrees. Thus, we have a right triangle 30-60. This is only half the equivalent triangle.

Second, what does this triangle tell us about the sides?

Now look at the right triangle 30-60. As an example, let pretend that the hypotenuse (the side opposite the right angle) has a length of 4. Remember that the hypotenuse is always the longest side in the right triangle. Knowing this length should also indicate the length of the short side, since it is exactly half the original side or 2.

Note. The link between the short side and the hypotenuse is always the same and is expressed as and 2a , Put these tags on a triangle.

Now we have a right triangle with two known sides. How to find a third party? You're right! We use the Pythagorean theorem: c ^ 2 = a ^ 2 + b ^ 2. For our example, this becomes 4 ^ 2 = 2 ^ 2 + b ^ 2 or 16 = 4 + b ^ 2 or 12 = b ^ 2. So, b = sqrt12 = sort (4x3) = 2sqrt3. Thus, the three sides, from the shortest to the longest, are 2, 2 square.

If you need more evidence, make some more examples to make sure that the ratio of the three sides of the right triangle is 30-60 ALWAYS a: a sqrt3: 2a.

Third: How do we use this relationship?

You must remember this relationship so that you do not have to repeat the Pythagorean theorem. You also need to understand exactly which side. always short side and 2a is hypotenuse, therefore sqrt3 this is another leg. Thus, if we know that we have a right triangle 30-60, then we need to know only one side in order to also know the two other sides, using this connection.

Example 1: In the right triangle 30-60 with hypotenuse 12. Find the other two sides.

Decision. Using a: a sqrt3: 2a relationship, and knowing that side 2a is equal to 12, means that the other sides 6 and 6sqrt3

In addition to the situation of finding the missing sides, this relationship can also be used to define the triangle as rights 30-60 or not. The sides of 5, 2.5, 2.5 sqrt3 correspond to our required ratio, therefore the triangle should be 30-60 right triangle.

In conclusion, knowing this connection and finding out when you can use it, eliminates the need to use the Pythagorean theorem. This is a definite short stretch!