My Fourth & Final blog...

The question that I have been investigating: Is there any general patterns in finding the side lengths of a right triangle?





In the right triangles that I have been working with, there seems to not be a constant equation that you can use in every right triangle. Each triangle has a different equation used, that is able to work everytime. For example, in my previous blog enty I had discovered an equation to get the Short Leg and the Hypotenuse Leg by using the Longer Leg, in a 10-80-90 triangle. Short Leg= Longer Leg x .1763 and Hypotenuse Leg= Longer Leg x 1.0154. Knowing that you are able to figure out the other two legs just by having one side length and angle measurements of 10-80-90 triangle is extremely cool! This pattern could be like the 30-60-90 triangle pattern... The next question in my investigation would have been: If I did tables of many right triangles, would there be any similar pattern to the one I have already discovered? The table and picture of a 10-80-90 Triangle is posted below.

Longer Leg	1	2	4	5	6	7	8	9	10
Short Leg	0.1763	0.3526	0.5289	0.7053	0.8816	1.0579	1.2342	1.4106	1.5869
Hypotenuse Leg	1.0154	2.0308	3.0462	4.0617	5.0771	6.0925	7.1079	8.1234	9.1388

Relationships

Since the last time I blogged I was investigating questions from my teacher, now I have made up my own question....

In a right triangle, if you change the hypotenuse leg, what will happen to the longer leg and shorter leg?

This right triangle is a 10-80-90:

There is a relationship between the Longer Leg and the Short Leg and between the Longer Leg and the Hypotenuse Leg. The relationship between the Longer Leg and the Short Leg is the Longer Leg x .1763 = Short Leg. This relationship is shown in the table below. There is another relationship between the Longer Leg and the Hypotenuse Leg. The relationship is Longer Leg x 1.0154 = Hypotenuse Leg. This relationship is shown in the table below.

Hypotenuse Leg	1	2	3	4	5	6	7	8	9	10
Longer Leg	.9848	1.9696	2.9544	3.9392	4.9240	5.9088	6.8936	7.8784	8.8632	9.8480
Short Leg	.1736	.3472	.5209	.6945	.8682	1.0418	1.2155	1.3891	1.5628	1.7364

I had to take many steps to investigate this triangle. I used the website: http://www.saltire.com/applets/triangles/tri1s2a.htm to find my information. It also took me some complicated calculations to get this pattern. My next step will be to finish looking at all Right Triangles and seeing if they have a similar pattern to eachother.

triangle pattern 10-90-80

Since the last time I blogged, I had discovered a pattern in a 40-60-80 Triangle. Now I am working with a Right Triangle, 10-90-80. I have discovered a pattern in the relationship between the side lengths.

The angle combination that shows a similar pattern is the 10-90-80 Triangle. There is a relationship between the Longer Leg and the Short Leg and between the Longer Leg and the Hypotenuse Leg. The relationship between the Longer Leg and the Short Leg is the Longer Leg x .1763 = Short Leg. This relationship is shown in the table below. There is another relationship between the Longer Leg and the Hypotenuse Leg. The relationship is Longer Leg x 1.0154 = Hypotenuse Leg. This relationship is shown in the table below.

Longer Leg	1	2	4	5	6	7	8	9	10
Short Leg	0.1763	0.3526	0.5289	0.7053	0.8816	1.0579	1.2342	1.4106	1.5869	1.7632
Hypotenuse Leg	1.0154	2.0308	3.0462	4.0617	5.0771	6.0925	7.1079	8.1234	9.1388	10.1542

I had to take many steps to investigate this triangle. I used the website: http://www.saltire.com/applets/triangles/tri1s2a.htm to find my information. It also took me some complicated calculations to get this pattern. My next step will be to finish looking at all Right Triangles and seeing if they have a similar pattern to eachother.

A Different Triangle Pattern

We have discovered that there are patterns that appear in the lengths of right triangles with the angles 30-60-90 and with the angles 45-45-90. Are there any other angle combinations that show similar patterns?

The angle combination that shows a similar pattern is the 80-60-40 Triangle. There is a relationship between the Short Leg and the Hypotenuse Leg and between the Short Leg and the Longer Leg. The relationship between the Short Leg and the Hypotenuse Leg is the Short Leg x 1.532 = Hypotenuse Leg. This relationship is shown in the table below. There is another relationship between the Short Leg and the Longer Leg. The relationship is Short Leg x 1.347 = Longer Leg. This relationship is shown in the table below.

Short Leg	1	2	3	4	5	6	7	8	9	10
Hypotenuse Leg	1.532	3.065	4.597	6.129	7.661	9.193	10.725	12.257	13.789	15.321
Longer Leg	1.347	2.694	4.041	5.389	6.736	8.083	9.431	10.778	12.125	13.472

I had to take many steps to investigate this triangle. I used the website: http://www.saltire.com/applets/triangles/tri1s2a.htm to find my information. It also took me some complicated calculations to get this pattern.

We have discovered that there are patterns that appear in the lengths of right triangles with the angles 30-60-90 and with the angles 45-45-90. Are there any other angle combinations that show similar patterns?