Talk:Solving Triangles

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Response to Checklist

I've written comments. I think that the current order of the content is fine, and I'd suggest doing a bit of edits on the mirror section once Leah has made more progress on the snell's law page. We can put it up live before then, though, if you only have edits related to mirrors left. AnnaP 7/10 Thanks, Anna! Comments all adressed! Richard 7/11

Messages for Future

  • Since this page is the link between all of my other pages, there's definitely room for all other geometric triangular concepts that an link to this page. In my "Ways to Solve", I've already got a red link for Pythagorean Theorem and there could easily be more red links with special rights and such.

References and Footnotes

  • All necessary references are made.
  • I know this is a bit nit-picky, but in your "Forest1.jpg" image, is that actually a tiny Smokey the Bear photoshopped in? If it is, you might need to remove it... I'm not sure if Smokey's image is public domain (though I'm sure you could try to make a "public figure" argument!)
Plain, old, generic Ranger Rick reporting for duty! Richard 7/11


  • The motivation behind this page is relatively clear, and the Why It's Interesting shows that this can be easily applied and viewed from a useful/real-world perspective.

Quality and Structuring

  • The Basic Description outlines the basic set-up of a shadow/triangle problem and the MME explains variations, says why it works, and ways to solve with examples.
  • This page was made so that its helper pages would have something to help.
  • The page is intended to follow a logical order.

Integration of Images

  • There are lots of images on this page, and each one is labeled clearly and goes with text that pertains/explains it.
  • Your triangle under "more than shadows" is too small to read the labels well
  • Create a little image to show which sides are a, b and c under pythagorean theorem. Ditto the next two, though those can share a non-right triangle image.
Done-zo! Richard 7/11


  • Once again, this page links to the topics of triangles in geometry (and maybe could link to a section in Snell's Law, but that section is more involved in physics).

Examples, Calculations, Applications, Proofs

  • There are plenty of examples on the page (3 and then a real life application).
  • All of the proofs for the math on this page are deferred to the helper pages (Law of Sines, Law of Cosines).
  • Where a formula or equation is mentioned (ex. Pythagorean Theorem, LOC, LOS) the equation is put on the page so that readers don't have to click away to be reminded of the equation.
  • The applications in the Why It's Interesting (and even in the example problems) are intended to be fun and exciting. The Eratosthenes bit is intentionally left vague so that it is evident he used shadows to complete his calculations, not that he used the same math that s contained on this page.
  • for your ship example, I'd suggest making the steps for each exactly parallel. You skip steps for the white ship that you show for the black ship--I suggest showing all of them for both. Also, it might be nice to have the math for the white ship to the right instead of the left, so that the calculations line up with the image.
Hope it's better!!!! Richard 7/11

Mathematical Accuracy and precision of language

  • The math on this page is written with clear, concise steps like my other pages, and the math itself was checked over and revised.
  • Your knight is on a horse bigger than any used for riding in Europe at the time that there were knights and castles. Either that, or your knight is like 7 feet tall. Not worth fixing, but I thought I'd let you know :)
We tried to figure this out for like 4 days! You should've been here for that discussion! Richard 7/11
My knowledge of horses comes from my mother :) Most older European breeds would be no bigger than 4 feet tall to the back. Your average European man during that area wouldn't be unlikely to be much over 5'6" or so (even aristocrats had poor nutrition back then). Take half a bit more than half that height, and really, the highest his eyes would likely be is 7ish feet, not 8. In America, we breed horses big, and so movies have shown historically inaccurately large horses for a long time! There are stock horse breeds that have been huge for centuries, but they tend to be a bit slow for a knight to want to ride!


  • The text is in short paragraphs and has been viewed in several sized windows.
  • Important words are bolded. Some common words, like "hypotenuse", are mouse-overs so the reader can be reminded of what it means.
  • Hide/show reduces the length of the page and only keeps the interesting parts (the pictures) visible.
  • For most of the white space on this page, a picture went into its space.
  • Columns do a pretty good job of organizing the text and pictures on this page.
  • Hidden text appears fine.
  • Add it a bit more white space right before "sight problems"
Comment Addressed! Richard 7/11

I'm really excited about this page. I think that there are some cool things on this page that make me want to format/create my next page like this one. Though not a whole lot of math is contained on this page itself, it clearly says where it can be found while presenting the information in a way that is fun and interactive.

Richard 7/7

General Comments

Chris, 7.5.11

  • There are really two main topics to this piece: Solving Triangles and Light Problems. Some of your content refers only to one or the other:
    • 1. mirror problems often don't involve solving triangles;
    • 2. ladder problems don't relate to light
  • I love the opening image from Shadows and Fog and how you get going with a topic and pursue its many possibilities. The images you use are terrific and really help the reader understand the concepts. With its many strengths, the page is quite long and goes in a number of directions. I think you would benefit from either splitting this page into a page on solving triangles and a page on light or focusing on solving triangles involving light only.

I tend to think that it wouldn't make sense to split this page up. They all involve solving triangles, and this page already doesn't have a ton of math to stand on (though it does have plenty to link to). For now, I think I'm going to tweak the Mirror problems section a bit (I have a better picture/example in mind) and maybe link to Leah's Snell's Law so I can reduce some of the content about the properties of light. I'm also considering hiding some of the content in the Why It's Interesting to make the page seem less long. I'd love to discuss this more/I can ask for any suggestions at the meeting this afternoon. Richard 7/6

  • I noticed that the picture of the person's shadow in the grass is over 2MB. It's displayed as a small image on the page, so maybe it doesn't need to be so large? I also felt like this page has a lot of photos that convey the same thing (the triangle that has the angle of elevation).
[[User:Nordhr|Nordhr] 18:09 27 June 2011
fixed the image. Richard 7/5
  • Kate 14:21, 20 June 2011 (UTC):
  • The order things are in now makes sense to me
  • Your pictures are awesome!

Section-specific Comments

  • Kate 14:21, 20 June 2011 (UTC): About the caption - the indentation on the second paragraph doesn't match the first. Is that fixable?

I couldn't figure this out. Richard 6/20

  • *Rebecca 23:06, 30 June 2011 (UTC) Great opening sentences. I think you hit the key points and draw the reader in!
  • Just one question- is "How tall really is the ominous character?" a proper sentence? The "really" seems like it's in a weird spot to me... maybe "How tall is the ominous character really?" I'm not sure if that's even proper English actually... just thought I'd point it out.
Agreed. I'll fix something up. Richard 7/5

Basic Description

Kate 14:39, 20 June 2011 (UTC):

  • According to postulates for Congruent triangles, given three elements, the other three elements can always be determined as long as at least one side length is given.
Not actually accurate, right? Because of the ambiguous case? So say something else.
Comment Addressed Richard 7/6
  • angle measure that can be measured by swinging from the horizon.
Swinging what from the horizon, and where are you swinging it to?
Comment Addressed Richard 7/6

  • Ultimately, a shadow problem asks you to solve a triangle by providing only a few elements of the possible six total.
It's possible to read that sentence as meaning that in your answer you only provide a few elements, instead of the problem only giving you a few elements.

  • Rebecca 23:41, 30 June 2011 (UTC) Great first paragraph!! Also nice used of images and spacing of the paragraphs.. makes it easy to read.

A More Mathematical Explanation

Why Shadows?

Kate 14:41, 20 June 2011 (UTC):

  • First two paragraphs in this section overuse the word "shine"
Couldn't think of a better way to reword this and keep the same conciseness and meaning. Suggestions? Richard 7/6
  • You need to introduce the cue ball better. Say something like, "We can think of light as a cue ball" or "Light behaves the same was a cue ball does", instead of "Another example is…"
Comment Addressed Richard 7/6

While the cue ball example is understandable, billiard players break the approach=departure concept all the time by using spin on the cue ball. If you think this can be safely ignored, fine, but if not you could easily say "ignoring spin on the cue ball..." to clarify this.

*Rebecca 15:54, 1 July 2011 (UTC) "Light is not like a liquid: it does not fill the space in which it shines like liquid assumes the shape of any container it's in." should be "Light is not a liquid: it does not fill the space in which it shines or assume the shape of it's container."

  • However, I'm not even really sure if you need this sentence. It seems like you cover the basic idea when you say "Light waves travel forward in the same direction in which the light was shined."
  • I have a proposition for reorganizing your pool ball paragraph. It's a great analogy, but it seems like you're using the behavior of light to explain the movement of the pool ball, rather than the pool ball to explain light. I think the anagoly is still really helpful to the idea, but it's more logical to word things the opposite way. Especially since your first sentence seems like your introducing a "new example," I don't want people to think that the pool ball is a seperate problem they need to solve with the help of their knowledge of light rays. I'm not sure if I'm getting my point across... but anyway, my proposed rewording is the same as yours but everything is flipped:
"It may be helpful to think of light reflecting as similar to a cue ball bouncing off the wall of a pool table. Just like the way that the cue ball bounces off the wall, light bounces off of the mirror at exactly the same angle at which it hits the mirror. The beam of light has the same properties as the cue ball in this case: the angle of departure is the same as the angle of approach. This property will help with certain types of triangle problems, particularly those that involve mirrors." Feel free to ignore this suggestion too if you disagree!
Comment Addressed. This was really helpful! Richard 7/6

More than Just Shadows

Chris, 7.5.11

  • Rather than SAS, ladder problems more often involve H-L because:
    • 1. the right angle is assumed;
    • 2. the height of the ladder is known;
    • 3. the distance between the bottom of the ladder and the wall is easily measured.

Comment Addressed. I think this is better. Richard 7/6

Kate 15:21, 20 June 2011 (UTC):

  • Most of these problems are formatted as word problems, that is set up the problem in terms of some real life scenario.
Comment Addressed Richard 7/6
  • they involve the length from the wall to the base of the ladder, the fixed length of the ladder itself, and the enclosed angle of elevation to determine the height at which the ladder sits on the wall.
But don't you already have the right angle? So you'd have SASA. Wouldn't you get something that was either SAS using the right angle or that allowed you to do law of sines?
Comment Addressed. See Chris's comment above. Richard 7/6
  • the angle of vision is the same exact angle at which the second person looks the mirror.
Are you sure this is true? what if they were standing like this?
Mirror ex.gif
Discussed with Kate before Comment was Addressed Richard 7/6

*Rebecca 16:08, 2 July 2011 (UTC) I agree with Kate's comments, but I found this section very clear otherwise. Great work.

Ways to Solve Triangles

Kate 15:23, 20 June 2011 (UTC):

  • I think you're overusing "numerous" in this section.
Comment Addressed Richard 7/6
  • You might want to explicitly state that the Pyth Thm is only good when you've got a right triangle.
Comment Addressed Richard 7/6
  • According to postulates for Congruent triangles, the AAA configuration proves similarity in triangles, but there is no way to find the side lengths of a triangle.
I would say "there's no way to prove congruency", because you've just said that in that configuration you can't get side lengths.
Comment Addressed Richard 7/6

  • Rebecca 17:37, 6 July 2011 (UTC) Nice section. It's great how you can link to different pages for each bullet.

Example Triangle Problems

Kate 15:30, 20 June 2011 (UTC):

  • Pirate ship problem: To find the distance between the two ships, we can take the difference in length between the two ships.
I think you mean the difference in length between the bases of the two triangles.
Comment Addressed Richard 7/6
  • I think you should call them ships consistently, and not boats. Boats are smaller.
Comment Addressed Richard 7/6

Why It's Interesting

Kate 15:32, 20 June 2011 (UTC): calling it a "paradigm" twice in such a short section is a little much.

Comment Addressed Richard 7/6

Example: Sizing up Swarthmore

History: Eratosthenes and the Earth

  • Rebecca 17:41, 6 July 2011 (UTC) The end of this page is really great! Do you have any information about how accurate Eratosthenes' calculation was? I think it would be interesting to put that information.
Comment Addressed Richard 7/6

There's a bit more to this story. According to, there was a well in Syene that was known to have sunlight reach its bottom at the summer solstice, and this was the impetus for the measurement of the solar shadow on that day. Adding this detail might make the story a little more interesting.

Old Stuff for Richard

Notice how the triangle in the picture on the left does not sit on the ground representing the distance from the knight's horse to the castle. Instead the base of the triangle is suspended at the eye level of the knight as he rides the horse. In this example, the knight can use a protractor to measure the angle as he looks up at the damsel in distress. He knows how far he is from the castle, but needs to figure out how high the tower is so he can rescue the damsel.

This type of problem not only calls for the triangle to be solved but also some additional information. If you calculate the height of the triangle, you are calculating the length from the eye level of the knight to the top of the tower. To calculate the actual height of the tower, so the knight can save the damsel, the height from the ground to the eye level of the knight must be added to the height of the triangle. In this set-up, the triangle is translated up some given distance. This is a common occurrence in triangle problems.

In this form of a sight problem, some person stands above some level and looks down at two separate things. In the example at the right, the man looks down at a white-sailed ship and a black-sailed ship. Because he has to look down at two different angles to see each ship, two different right triangles are formed by his line of vision, the height of the cliff, and the distance between each ship and the base of the cliff.

A problem like this will often ask for the distance between the two ships. This calls for both triangles to be solved, since the distance between the two ships is the difference between the lengths of the bases of the triangles.

Ships sailing.jpg