Side && Angles 3

Using the table from my previous blog entry, I will answer the following questions:

1. So what does this make you think about triangles?
2. Do you have any more questions?
3. Can you make any generalizations?

With the information that I gathered using the table, it tells me that when all the sides of a triangle are increased by the same number nothing really changes because the sides stay proportional. It also tells me that even though two triangles have different sides it is possible that they can still have the same angle measurements. I also learned from Ms. Sheppard-Brick that there is a long list of triangles that involve other parts of the triangle, as I predicted in an earlier entry. Although I did figure that out myself before doing my own investigation, the table (list) is called Trigonometric Functions.
Some questions that I have are:

• What does the chart with the triangles, sides & angles show?
• How did mathematicians figure this out with out an equation?

A generalization that I can make is that if any three sides of a right triangle are increased by the same number multiple times they will continue to have the same angle measurements each time. A few more patterns that I noticed that all the numbers in the previous table are even numbers. I doubt that, that makes a difference because there are odd numbered sides that make right triangles too, and I believe if those stay proportional the angles will be the same as they started out.

Side && Angles 2

Side:	Angles of w/o 90:	Multiplied by:
6,8,10	36.9 & 53.1	-
18,24,30	36.9 & 53.1	3
54,72,90	36.9 & 53.1	3
12,16,20	36.9 & 53.1	2
24,32,40	36.9 & 53.1	2

After I noticed, and was told, that there are triangles that have different side lengths but the same angle measurements. I also noticed that the numbers increased by 3, with the triangles from my previous blog. I wanted to see if that was a just a coincidence that they had the same amgle measurements so I tried multipling the sides in the 18-24-30 triangle by three then those numbers that I got also by three, and I still ended up with the same angles measurements. Then I tired multiplying the 6,8,1o angle by two and then, again, the those side lengths by two and still for both set of sides I got the same angle measurements.

Sides & Angles.

When investigating if it is possible to figure out the angles of a triangle by knowing the side lengths, found something that changed my idea. The numbers 6-8-10 are a pythagorean triple. the numbers 24-18-30 are all multiples of 3, 6 multiplied by 3 equals 18, 8 multiplied by 3 equals 24 and 10 multiplied by 3 equals 30. Which is a pattern I found interesting. Another thing that I noticed s that when I used 6-8-10 and 24-18-30 they had the same angle measurements. If there is a specific way to figure out how to find the angles of a triangle when you know their side lengths, how can you find thatwhen you can have different side lengths give you the same angle measurement?

I think that there are other parts of triangles that would help us determine the angles in a triangle, when you only know the side lengths of that triangle. In geometry for the triangle unit we have only been working with the angles and sides of a triangle but there has to be many other parts of a triangle that could help us figure out the measurements of the angles. Based on the information we have been learning we know how to find the angles of a triangle when there are two triangles involved and an angle is already given, which does not seem to be enough to help us figure out the measurements of the angles in a triangle.

Relationship between sides & angles of a RIGHT triangle - Continued

During the previous investigation on figuring out the angles of a right triangle if you know their side lengths, some very intelligent people brought up a few questions.

1. Do you see any pattern in the way that the numbers change?
2. What kind of data will be most helpful? How do you think it will be more helpful to organize your data?
3. Do you think it is possible to predict the angles of a triangle when you know their side lengths?

1.By looking at the chart from my previous blog, I saw a pattern in the way that Angle B changed in the first set of sides and angles, before the break. The pattern that I saw is that the measurement of Angle B increased by 4, not exactly 4 but close to it. The pattern does not continue throughout the entire table, but continues for most of the table except the beginning and the end.

2. The data table was helpful for the fact that it put all my information next to each other so that I could see it better. I think a better way to organize this information by still using the chart is keep all the parts that are changing in one group, so I can see how they interact. This will be more helpful because the other information around makes it harder to focus on that one part of the triangle.

3. I do think that it is possible to figure out the angles of a triangle when you only know the side lengths. Maybe not by making a chart with the sides and angles they make, like I did. But possibly if I knew more things about triangles it would be easier. For example I know that if two triangles have the same side lengths they have the same angle measurements too, due to the Side-Side-Side Congruence Shortcut that I learned. Which basically says if I have a triangle that have the exact same lengths they will also have the same size angles. So there most likely is a way to figure out the angles of a triangle by knowing the side lengths.

Relationship between sides & angles of a RIGHT triangle.

If you know the side lengths of a right triangle, can you predict what the angles will be?

. Side AC: 9 9 9 9 9 9 . Side BC: 9 8 7 6 5 4 . Hypotenuse AB: 12 12.042 11.402 10.817 10.296 9.849 . Angle A: 45 41.63 37.88 33.69 29.05 23.96 . Angle B: 45 48.37 52.13 56.31 60.95 66.04 . . Side AC: 9.5 8.5 7.5 6.5 5.5 4.5 . Side AB: 9 9 9 9 9 9 . Hypontenuse BC: 13.086 12.379 11.715 11.102 10.548 10.062 . Angle :B 46.548 43.363 39.806 35.838 31.43 26.565 . Angle C: 43.452 46.636 50.194 54.162 58.57 63.435 . . Side AC: 8 8 8 8 8 8 . Side BC: 9 8 7 6 5 4 . Hypotenuse AB: 12.042 11.314 10.63 10 9.434 8.944 . Angle A: 48.37 45 41.19 36.87 32.01 26.57 . Angle B: 41.63 45 48.81 53.13 57.99 63.44

This chart shows the relationship between the changing of the sides and the angles. One pattern that I noticed is that when you change the size of side AC which is the longer side of the triangle, Angle A decreases in it's size, however the size of Angle B increases. This pattern also works when you change the length of side BC the shorter side of the triangle. Just as the measurement of Angle A, the size of the hypotenuse decreases.