Length of Diagonal

[jedishrfu]

This is a common theme with teachers to remind students to never trust a diagram and eyeball an answer from it even though in the real world one might do that. But students being students will try to optimize their energy in such a pursuit.

I have been known to do that and then I learned to listen for clues from the teacher like this looks like a great quiz question, or subtlety when I was a student I got tricked by this. It's helped me on three separate occasions in college and a few times in high school too.

 

[SammyS]

You see I can draw this triangle like this. It's nothing more than a typical mathematical construct.

Take for example this triangle

The objective is to find $b$ and $\theta$. So most obviously you reach for the Law of Cosines for $b$, and the law of sines for $\theta$.
. . .

No, that's not so obvious.

You're right about first determining $b$, and the Law of Cosines will do that for you.
But, if next, you're going to use the Law of Sines (followed by application of the arcsine function) to determine one of the remaining angles, you would be well advised to pick an acute angle. That angle is the unmarked angle in your figure, the angle opposite the side of length 42 . This angle is guaranteed to be less than $90^\circ$. The largest angle is opposite the longest side, and no more than one of the angles of a triangle can be obtuse.

In short: Avoid using the arcsine to solve for the largest angle.

 

[jedishrfu]

There was someone on PF who posted a trick interview question with an isoscelles triangle with each being 10m and the base being 20m asking for the angles.

It came with a triangle diagram labeled as such causing some confusion since given the side values its becomes clear its not a triangle when two sides sum equals the third side.

Who was that masked poster?

 

[Herman Trivilino]

I don't want the option to participate.

Yet you continue to participate. Grow up.

 

[brotherbobby]

You said @erobz,
"I've drawn a sloppy mathematical diagram of a square of side length a as I and almost everyone does".

No I dont. To draw a neat diagram is essential for thought, unless the problem is (trivially) simple. I go above and beyond to do it, or else I can't think. I suppose it will be less inportant for others. Yet, a diagram inaccurately drawn can make you suspect your own thinking. You could be correct and believe you aren't.

I insist on all students drawing accurate diagrams to questions, when lengths and angles are given. They lose a little mark if they do not.

 

[FactChecker]

You can at least make the 42 side shorter than the 56 side can't you? That doesn't seem to be expecting too much.

 

[erobz]

Let's debate on principal. Visually examine the triangle...Is it drawn incorrectly?

 

[erobz]

You said @erobz,
"I've drawn a sloppy mathematical diagram of a square of side length a as I and almost everyone does".

No I dont. To draw a neat diagram is essential for thought, unless the problem is (trivially) simple. I go above and beyond to do it, or else I can't think. I suppose it will be less inportant for others. Yet, a diagram inaccurately drawn can make you suspect your own thinking. You could be correct and believe you aren't.

I insist on all students drawing accurate diagrams to questions, when lengths and angles are given. They lose a little mark if they do not.

But it shouldn't be essential for the mathematics. This isn't about "marks"

 

[DaveC426913]

Let's debate on principal. Visually examine the triangle...Is it drawn incorrectly?

View attachment 359560

I'd say an inaccurate diagram serves the purpose of teaching students to "do the math" and not trust diagrams.
But in a real world scenario, an accurate-(ish) diagram is a good form of sanity check.

Somewhere out there on the webz is a diagram of a triangle that is actually a degenerate straight line (altitude=0) but this fact is obscured by a diagram of an innocent-looking typical triangle.

 

[Mark44]

Let's debate on principal. Visually examine the triangle...Is it drawn incorrectly?

It's an order of magnitude better than the one you drew in post #6.

But it shouldn't be essential for the mathematics.

A human brain has two hemispheres. As I understand things, one side is more visually oriented and the other side is more analytically oriented. An inaccurate drawing is a hindrance in doing the mathematics, something that others in this thread have noted.

 

[erobz]

It's an order of magnitude better than the one you drew in post #6.

You haven't answered. Is it incorrectly drawn?

 

[Mark44]

You haven't answered. Is it incorrectly drawn?

It looks fine. My comment was more about your drawing of post #6, which was obviously drawn incorrectly.

 

[erobz]

It looks fine.

stay on post 37 for a moment... But its clearly incorrectly drawn - not fine. So I disagree. An accurate diagram would capture both triangles that satisfy the constraints.

 

[Mark44]

stay on post 37 for a moment... But its clearly incorrectly drawn - not fine. So I disagree. An accurate diagram would capture both triangles that satisfy the constraints.

I see nothing wrong with the triangle you drew -- you drew one triangle, and the sides you labeled are not clearly out of whack, unlike your figure of post #6.
The fact that there might be two distinct triangles with sides and one angle as shown is a consequence of the information given with the picture; namely, two sides and a non-included angle. It is very well-known that such triangles are not unique.

You're beating a dead horse here. Please stop.

 

[erobz]

I see nothing wrong with the triangle you drew -- you drew one triangle, and the sides you labeled are not clearly out of whack, unlike your figure of post #6.
The fact that there might be two distinct triangles with sides and one angle as shown is a consequence of the information given with the picture; namely, two sides and a non-included angle. It is very well-known that such triangles are not unique.

So you check the ambiguity criteria's every time you use the law of sines to see which triangle you need...what happened; you said, "it was fine"? You had no idea from the diagram that it wasn't unique. The Law of cosines did.

 

[Mark44]

So you check the ambiguity criteria's every time you use the law of sines to see which triangle you need...what happened; you said, "it was fine"?

I have retracted part of what I wrote, both in my post and in what you quoted.
First off, the drawing is fine. Second, no, I don't check ambiguity criteria (BTW criteria already is the plural of criterion).

I would use the Law of Sines to find the angle across from the $36.3^\circ$ angle and would then find the remaining angle. This is the same advice @SammyS gave some time ago.

Since you are bringing nothing new to this thread, I'm labeling you done in it, again.