@micheal-daubyn-2
???? I think we all have a hundred or so similar things stashed around. Here's a picture of a solution done in Google sheets. It's all law of sines except I used law of cosines for the check on the final segment.
This one was done in GeoGebra, which is a math teacher's big-stick calculator.?ÿ
It's just amazing how far we've come with calculating and graphing instruments. With the spreadsheet, you have to know some trig, but there's no look-up, just write the formulas. With GeoGebra, it's all drawing selections and settings.?ÿ
I've always thought that I lived through a great period. All of my math learning was without any machine help and a good bit of the application has been with electronic help.
I wish we could teach everything all ways now, but there's just not time. I think we're getting a slice in the middle, leaving out some of the important supporting stuff and not going far enough with the computers.
?ÿ
I know exactly what you mean.
In my surveying course, we had to do a least squares resection "by hand" armed with nothing more than a pencil and a sheet of paper. We had to calculate all the way through to the residuals stage as well.
Fast forward 15 years, and I went back to where I studied and spoke to the head teacher. We talked about this massive long-winded equation. He told me that the kids these days only need to learn how the crunch through the data on their HPs with a program called Quickclose.
I remember being angry on the way home. I was upset that these kids are running around with same qualification as me.
I can't for the life of me remember the equation off the top of my head today, but as a direct result of having the equation explained to me step by step, I can see it in my mind. I understand what the residuals represent because I can actually see it.
These kids are walking blindly through a world of numbers.
Not cool.
That only works when the hypotenuse is "equal" on both triangles.
@micheal-daubyn-2
I learned to do everything using DMD sheets and when I took my state exam, I showed my work on hand-drawn DMD sheets.
I think it's tighter than that. The two triangles have to be congruent. If their hypotenuses (hypotenii?) are equal, then either a leg from one must also be equal to a leg from the other or an angle of one must also be equal to an angle of the other to guarantee congruence.
You've created a great example of a statement and its converse. Statement: If it works, then the hypotenuses are equal. Converse: If the hypotenuses are equal, then it works. The statement is true but the converse is sometimes false, so the converse is not true.
That's one I would use in class if I still had classes, but I'll pass it on to some real teachers. Thanks for pointing it out!
@micheal-daubyn-2
Sounds like you went to OIT.
It all depends upon what resulting shape you will have from a square to an infinite reaching spear.
?ÿ
Almost. Consider the diagram below:
Both triangles ABC and ABD are 3-4-5 right triangles. Note that they form a quadrilateral consistent with the "wrong triangle" description; four right angles.
Now let's move point D along the circumscribed circle. Note that both triangles are still right triangles, because they're both inscribed in semi-circles.
The two right triangles share the hypotenuse, so their hypotenuses are equal. However, the figure no longer fits the "wrong triangle" description because it has only two right angles. Triangle ABD is no longer a 3-4-5 right triangle, but it is a right triangle and its hypotenuse is the same as the hypotenuse in triangle ABC.
Thus the condition that the hypotenuses be equal is a necessary condition for the figure to work, but it is not a sufficient condition. For sufficiency, the triangles need to be congruent, and that requires another condition in addition to the hypotenuses being equal.
?ÿ
?ÿ
Congruent simply means that the triangles are equal, that all sides and all interior angles are the same for each triangle.
The worst situation is when they have equal angles and different length sides, which would be called similar triangles and have been the cause of many arguments during the discussion of triangles.
Basically, They do not fit together and get confused for congruent triangles.
I was taught about right triangles and equilateral triangles and scalene triangles.
The only wrong triangles I know of is when your computations claim there is not 180?ø in the total of the three included angles.
Perfect pairing for T2 and HP3805 traversing.?ÿ
You're really still using this equipment? Traversing for what kind of projects?
I got about 99% of the way through writing a triangle solver app for Android.
What's left to do?
You raise good questions. The whole idea of left triangles and wrong triangles was just a play on words. One of the hard transitions to geometry for students who were super good in algebra was how important the precise meaning of words is in geometry. It always amazed me the number of good algebra students who stumbled in geometry and the number of poor algebra students who thrived in geometry. Sometimes we would joke in class?ÿ about such undefined concepts just to make the point.
Similarity was a difficult concept as well. We relax a requirement for congruence and get the concept of similarity. From that we can get an idea of trig functions, although that's not the way they were originally conceived.
One two-dimensional "wrong triangle" concept is a triangle whose angles sum to more than 180 degrees, but we can imagine another one based on the sides. For example, make a triangle whose sides are 5, 8, and 14 using units of your choice.
An algebraic concept that the smartest guy I ever taught with and I used to argue was this: Is the repeating decimal 0.999999... exactly equal to 1? That concept sometimes comes up in computer algorithms, usually in terms of when a computed result is really zero. I had it right (of course!) but he was and is way smarter than I could ever hope to be.
I love ol' stuff like this.
?ÿ
?ÿ
?ÿ
?ÿ
?ÿ
?ÿ
?ÿ
special projects, not typical production. Mostly because the gear works as well as the day it left the factory. Just feels right to shoot 10,000 ft with EDM, and 1? angles the hard way.?ÿ
Probably not much. It was part of a much bigger surveying app for android, and I eventually scrapped it for spare parts because life went in a different direction.
If I remember correctly, the triangle solver section of the app was in the testing stage.
I published another section of the app as a stand-alone app (a survey-specific unit converter), but was unhappy with the way the Google Play Store published it.
The triangle solver was fully reciprocating. Enter any 3 known variables (the angles could be any format. The app would convert them) and the app would solve the whole triangle. It would also give both answers if it was an ambiguous case triangle, and show you a little moving graphic showing you why it was an ambiguous case triangle. It would also give you all corresponding circle info too.
I also wrote my own keyboard code as well, so those of us with fat fingers could use the app. My keyboard replaced the standard android keyboard.
We had a Leica Wild 1800 and the 2003 sitting in a store room with a NA2002 and NA2003 and traverse prism set.?ÿ When I asked the upper echelon what we did with them he just shrugged his shoulders and asked if I could try to sell them on Ebay.?ÿ He wasn't into surveying anymore.
