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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  • 390
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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  • 345
22 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 ...
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  • 7,703
18 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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  • 131
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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  • 231
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

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 ...
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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 $AC$ ...
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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

A formula for the area of a rectangle

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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 ...
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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 [...] ...
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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-...
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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 ...
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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 ...
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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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  • 6,377
3 votes

Visual representation of Cartesian Product of groups

Perhaps this YouTube video on Visualizing Group Theory, by Nathan Carter (Bentley Univ), may help. Here is a snapshot illustrating $C_3 \times C_4$:           Blue ...
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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 ...
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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 ...
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  • 2,028
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.
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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. ...
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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

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 ...
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2 votes

The Fundamental Theorem of Calculus and Vegetables

I remember this anecdotal version, I don't know whether or not it matches yours. The Fundamental Theorem of Calculus stated using vegetables: $$\int_{carrot}^{potato}vegetable(turnip)d(turnip)$$ $$=...
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  • 187
2 votes

Visual representation of Cartesian Product of groups

to illustrate $A \times B,$ draw two orthogonal axis and mark the points of $A$ on one and the points of $B$ on the other. the elements of $A \times B$ are the points where lines through points on the ...
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  • 241
2 votes

About visual ways of teaching Math

For complex variables, there is the famous "visual" book Tristan Needham, Visual Complex Analysis
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  • 6,377
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 ...
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  • 3,765
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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