Re: [Phys-L] area of the shaded triangle (John Denker)

[Scott Orshan <sdorshan@aol.com>]

Interesting. When I first saw the problem, it said that the outer box 
was a square, as in 
https://mindyourdecisions.com/blog/2018/04/26/what-fraction-is-shaded-maths-problem-stumping-the-internet/

So I started by identifying as many angles as I could. That was fairly 
useless.

As far as the general solution, for any rectangle, just looking at it, 
it seems as if the area will go from 1/2 to 1/4 of the area, as the 
slanted line sweeps from the upper left corner to the upper right 
corner. That's a good way to check the general answer, as it must work 
in the corners and at the midpoint.

Spoiler solution follows.




The equation for the fractional area of the shaded triangle, as the line 
moves from the upper left corner (call this x=0), to the upper right 
corner (call this x=1) is 1/(2x+2). At the midpoint, where x = 1/2, that 
gives the answer of 1/3. At the upper left, it is 1/2. At the upper 
right, it is 1/4.

It's easy(*) to work out if you assume a 1x1 square with an area of 1. 
The slanted line creates two similar triangles. The base of the big one 
is 1, and the base of the small one is x. If the lower one has height h, 
the upper one has height xh, and they add up to the full height of 1. 
h+xh=1, so h=1/(x+1). The area of the triangle in question is 1/(2x+2), 
which is also the fraction, since we chose a starting area of 1 for the 
whole thing.

(*)I did not reach this answer in as straightforward a method as it 
sounds. I had to draw a lot of pictures and go down a few different 
paths, before the solution became apparent.

Scott Orshan
Engineering Teacher
Bound Brook High School

On 4/29/2018 12:00 PM, phys-l-request@mail.phys-l.org wrote:
> Message: 1
> Date: Sun, 29 Apr 2018 06:37:27 -0700
> From: John Denker <jsd@av8n.com>
> To: Forum for Physics Educators <Phys-L@Phys-L.org>
> Subject: [Phys-L] area of the shaded triangle
> Message-ID: <b972c6b1-a2bb-efaa-c05f-7da4b2adaf5c@av8n.com>
> Content-Type: text/plain; charset=utf-8
>
> The following puzzle is somewhat amusing.
> It went viral recently:
>    https://amp.businessinsider.com/images/5ae3808042e1cc5185248f88-750-563.jpg
>
> It might be useful as a start-of-term pretest, to see
> if students actually learned anything in HS math.
>
> Note:  The steeply sloping line meets the *midpoint* of
> the top edge.  I reckon the tick marks in the diagram
> are supposed to indicate this, but it's a better puzzle
> if solving it does not require knowing this obscure bit
> of notation.
>
> ===============
>
> Would anybody care to comment on the following criteria
> for judging answers:
>
>   -- Some traction (as opposed to just giving up).
>   -- Numerical solution, but not correct.
>   -- Correct numerical solution.
>   -- Simple, elegant, insightful solution.
>
> I would argue that the above leaves out three things,
> all somewhat related.  Hint/giveaway:
>
> Fgneg ol bognvavat hccre naq ybjre obhaqf.
> Purpx gur jbex hfvat obhaqf be bgurejvfr.
> N erny zngurzngvpvna jbhyq sbezhyngr gura
> fbyir gur trarenyvmrq ceboyrz, abg whfg
> gur zvqcbvag pnfr.
>
> Meta-hint: http://www.rot13.com/
>
>
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