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What Is The Length? 99% Fail – Mind Your Decisions
I covered these problems before in 2019 in separate posts. But I felt it was worth covering in a single video to illustrate an important lesson.
Here’s a fun little puzzle. What’s the length of the top side?
When I shared this puzzle, most people were able to find the correct answer. You might say that 99 percent could find the answer.
But let’s make one change to the puzzle, which I credit to a comment by Michael Timothy. Change the left length from 4 to 3, and now see if people can solve for the top side.
So what do you think? What is the value of x in this shape? Incredibly many people fail to get the correct answer. So what is the correct answer?
As usual, watch the video for a solution.
Or keep reading.
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Answer To
(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).
Let’s tackle the first problem.
So how do many people solve this? They construct the hypotenuse of the 3-4 right triangle, which will have a length of 5. And then they will construct a vertical line, saying it has a distance of 3. So they get another 3-4-5 right triangle, and a shape like this:
If this figure were accurate, then the value of the top side would be 5 + 3 = 8. And this is the correct answer!
However, there are a couple of unjustified steps in this derivation. You would need to show the horizontal 5 is perpendicular to the vertical line, and you would also need to show the length of 3 reaches the endpoint.
So what’s the danger of using this method, if the answer was correct anyway?
The problem is this derivation will not work in general. Let’s see what happens when we use it to solve problem 2.
By this same method, the top side should be 5 + 4 = 9. But this is not the correct answer!
The construction makes a couple of mistakes: the blue line at the bottom is not parallel to the top line, and the vertical line marked 3 is not actually a distance of 3.
The point is made clear when the diagram is re-drawn to scale.
I find it interesting that students are basing their reasoning on the diagrams themselves.
In my math education, we were always taught never to assume diagrams are drawn to scale, and we were also taught not to assume angles like perpendicular or parallel lines.
Teachers are not being sinister by presenting diagrams not drawn to scale. Diagrams are meant to show relations in the big picture. In real life many diagrams are not drawn to scale: think about schematic maps of subway lines, atomic models, solar models, or maps of countries of the world. You should not assume a diagram is drawn to scale!
Solution to problem 2
My technique to solve this problem is to consider the distance from the bottom left corner to the top right corner.
This works out to √(42 + (3 + 5)2) = √80, as it is the same distance of the hypotenuse of a right triangle with legs of 4 and 8.
Now go back to the original figure.
And then consider just the right triangle as shown.
We can then readily solve for x using the distance formula:
x2 = (√80)2 – 32
x2 = 71
Since the side length is a positive number, we solve that x = √71.
In geometry you cannot always trust the diagrams, so the problem is a little bit trickier than it seems at first!
Source
Comment from Michael Timothy
https://www.youtube.com/watch?v=amLDMD8KPzQ&lc=Ugw4bQRoE0tit8sLNPR4AaABAg.8xArwe3DbZu8xCMU8Idv0-
Presh, please redo this problem using the bad solution given below, based on 3-4-5 triangles. Then do that proof again, except make the far left 4 a 3. That solution gives the answer ‘9’, while your correct method here gives root(71).
solar system nasa not scale size
https://commons.wikimedia.org/wiki/File:Solar_System_graphic_by_NASA.png
https://en.wikipedia.org/wiki/File:Scale_of_the_Solar_System.png
original video
https://www.youtube.com/watch?v=kfijuP5HDjU
Jakub Nowosad, CC BY-SA 4.0
https://en.wikipedia.org/wiki/Harry_Beck
https://en.wikipedia.org/wiki/Harry_Beck#/media/File:Tube_map_1908-2.jpg
https://commons.wikimedia.org/wiki/File:Harry_Beck%27s_tube_map_at_Finchley_central_tube_station_-_geograph.org.uk_-_6206154.jpg
Harry Beck’s tube map at Finchley central tube station by Mike Quinn, CC BY-SA 2.0
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Mind Your Decisions is a compilation of 5 books:
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