23
votes
Given a 3 4 5 triangle, how do you know that it is a right triangle?
The fact that there is a 3-4-5 triangle that is a right triangle is unique to the Euclidean plane. There is no such triangle in the spherical or hyperbolic planes. Since the Pythagorean theorem is ...
- 10.3k
23
votes
Given a 3 4 5 triangle, how do you know that it is a right triangle?
The Chinese came up with the following a long time ago. Probably something better, but this is the gist of it.
Let's start with a right triangle with height b=4 and base a=3. We know it has some ...
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22
votes
Given a 3 4 5 triangle, how do you know that it is a right triangle?
Here is a simple construction. Adjust to taste.
1. Draw a line $l$ passing through a point O.
2. Construct circles of radius 3 and 4 with centre O. Call them $C_3$ and $C_4$. Let the intersection of ...
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20
votes
Given a 3 4 5 triangle, how do you know that it is a right triangle?
Graph paper (or square floor tiling) to the rescue!
Proof by picture for a 3 4 5 triangle:
Because the drawing is on the grid and not the skew tiling of the square on the hypotenuse, determining ...
13
votes
Given a 3 4 5 triangle, how do you know that it is a right triangle?
Diagram shows that there exists a 345 triangle that is right-angled.
It is clear by inspection that an angle greater than 90 between 3 and 4 leads to a hypotenuse longer than 5. Similarly an angle ...
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13
votes
Given a 3 4 5 triangle, how do you know that it is a right triangle?
A complete different approach reasoning with the area $A$ of the triangle 3-4-5.
Use Heron's Formula to show that $A =6$.
Conclude that the height to the side with length 4 must be 3 since $A = ah/2$....
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10
votes
Accepted
A formula for the area of a rectangle
First copy your rectangle like this to make a big square of side-length m with a square of side-length d drawn inside it.
The big square minus the small square leaves four half-rectangles (coloured ...
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6
votes
Given a 3 4 5 triangle, how do you know that it is a right triangle?
I don't think I can do better than Giles answer, but here is an answer which gets the converse of PT without proving PT first:
Let $AB=3$, $BC=4$ and $AC=5$. Draw a point $D$ on the line segment $AC$ ...
- 5,243
6
votes
Styles of visualization in geometry
It seems the Stanford Encyclopedia of Philosophy has a useful article that at least
tangentially addresses your interesting question:
"Can visual thinking lead to discovery of an idea for a proof ...
- 29.5k
4
votes
Given a 3 4 5 triangle, how do you know that it is a right triangle?
If you have Pick's formula at your disposal, you can draw your favourite right triangle on grid paper and count. Actually, you can do the counting for any triangle with grid point vertices, but of ...
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4
votes
Given a 3 4 5 triangle, how do you know that it is a right triangle?
According to the Wikipedia entry for Pythagorean theorem, a proof of the converse of the Pythagorean theorem without assuming the Pythagorean theorem can be found in Stephen Casey, "The converse of ...
- 10.8k
4
votes
Given a 3 4 5 triangle, how do you know that it is a right triangle?
Not an answer; rather an observation.
If you can convince that the blue circle of radius $3$, and the black circle of
radius $4$,
meet the red circle of radius $\frac{5}{2}$ at the same point, then ...
- 29.5k
4
votes
Visual Pythagorean demonstration
Maybe you can do President Garfield's proof and combine it with a history lesson.
page 161 of the New-England Journal of Education, April 1, 1876 (image from Google Books)
Note: M. C. = Member of ...
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4
votes
4
votes
Styles of visualization in geometry
You might find the essay by Marjorie Senechal, "Visualization and visual thinking." Geometry's Future (1991): 15-21, published by the Consortium for Mathematics (COMAP), of interest. In particular she ...
- 1,872
4
votes
The role of visualization and intuition in graduate and postgraduate math and developing it
First, I am a great fan of Visual Complex Analysis.
Nevertheless, I disagree with the statement that
Mathematics today [...] is mostly built on abstract symbolic manipulation rather than on [...] ...
- 29.5k
4
votes
Are there mathematical proof info-graphics?
Math With Bad Drawing has some images that approach an info-graph (and in general is just a great website for math education), for example:
https://mathwithbaddrawings.com/2015/07/01/infinity-plus-...
- 1,941
3
votes
Visual aids for understanding group theory
John Jones has a great visualization tool for a nice selection of finite group tables. It allows you to do things like select a subgroup, and see the group table colored according to the cosets. ...
- 24.4k
3
votes
Visual aids for understanding group theory
The book Visual Group Theory by Nathan Carter seems to be a rich source of the kind of materials that you are looking for.
- 1,715
3
votes
Are there mathematical proof info-graphics?
"Kids usually struggle with every one of these concepts, let alone all of them together. It is difficult to get the whole picture and all the moving parts. So this place (proof) seems like a good ...
- 2,075
3
votes
Accepted
The Fundamental Theorem of Calculus and Vegetables
I am grateful to Mark Conger for finding a video of this presentation and getting the University of Michigan to digitize it. It isn't produced in the style of math youtube content, because that was ...
- 21.2k
3
votes
Plainly by eye, how can 16 year olds visually distinguish $\color{red}{\vec{b} - \vec{r}}$ from $\color{dodgerblue}{|\vec{b}| - |\vec{r}|}$?
No one diagram is going to help every student actually absorb this idea. In fact, the mistake reminds me a bit of students thinking $\sqrt{a^{2}+b^{2}}=a+b$. They are "simplifying" in a way ...
- 20.1k
2
votes
Given a 3 4 5 triangle, how do you know that it is a right triangle?
Cut out four identical 345 triangles and put the biggest angle from each triangle together at a point. Then observe that all 4 identical angles fit to make a revolution, which implies each of the ...
- 3,875
2
votes
Visual Pythagorean demonstration
This was in my math folder, I don't know its source.
It was shown as 'not needing any further explanation', but of course, for lower level students, I connect the 3 points on the circle's perimeter ...
2
votes
About visual ways of teaching Math
The slides of Dror Bar-Natan are excellent examples of such mathematical "infographics".
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2
votes
Given a 3 4 5 triangle, how do you know that it is a right triangle?
If you want to convince someone that certain triples are the side lengths of a right triangle, you can exhibit pictures like the following, which show the $(4,3,5)$, $(12,5,13)$, and $(8,15,17)$ right ...
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2
votes
What would you recommend for the math thinking course for school?
Have them work through a few chunks of Euclid's Elements. Then they are doing proof, rather than reading about logic in a theoretical way, which is probably too abstract for kids in the age group you'...
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2
votes
Accepted
Plainly by eye, how can 16 year olds visually distinguish $\color{red}{\vec{b} - \vec{r}}$ from $\color{dodgerblue}{|\vec{b}| - |\vec{r}|}$?
I would tell students that absolute value bars represent distance, and distance is indicated by a line segment without an arrow.
This rule of thumb is very intuitive:
students should already know ...
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1
vote
Plainly by eye, how can 16 year olds visually distinguish $\color{red}{\vec{b} - \vec{r}}$ from $\color{dodgerblue}{|\vec{b}| - |\vec{r}|}$?
My first reaction is to doubt the problem is that they somehow can't see the difference (how could you miss the vertical lines?), but rather that labeling only one vector with its length is throwing ...
- 1,527
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