Thursday 20 June 2013

Two ways to Solve a Geometry Problem

Before you read on, you might try to solve the problem above.

Greg Ross at the Futility Closet is a constant source of entertaining information and frequently posts some really nice mathematical problems. Recently he posted the one above. I wanted to reprint it because it dramatically illustrates how simple problems can be made complex by the the failure to use all our mathematical perceptions.  I have often conjectured that many students struggle with geometry because their vision is too literal.  They see what is there, only what is there, and nothing more.  And geometry, it seems, often asks us to see a line that isn't there, or a different orientation of the same problem, if we wish to see it at its most beautiful form.

I suppose the most difficult way to solve the problem would be to assign a unit length to the sides of one triangle in order to find its area.  Proceed to find the radius of the circle, and use that to establish a side of the larger triangle from which the area can be found. Please don't make me do that!

Probably a quick way for a geometry student with a little appreciation of the relationship between lengths and areas in similar triangles would be to establish that from the common center of the objects the distance to a leg was half the distance to the opposite vertex.  So the distance of a perpendicular segment from the common center to the parallel bases of the  two triangles was in a relation of 1:2,  allowing us to conclude the similar figures were in a ratio of 1:4.
As simple as that is, there is a calculating free method to solve the problem.  Simply invert the inner triangle as shown below and ..."Behold" as Brahmagupta said,
A familiar configuration that immediately shows the larger triangle is made up of four of the smaller.
That's how I WANT to learn to look at geometry problems.  Still working on it. 

I think I nice followup visualization challenge would be to ask the student to explain, without calculation, what would happen to the ratio of the two triangles if they were not equilateral and both remained similar and retained the circle as their respective incircle and excircle?


CCSSI Mathematics said...

Yep, looks familiar indeed:

But then, some problems require a bit more than just visualization skills:

Pat's Blog said...

Nice problems, don't know how I've missed your sight, but you've won a fan here.
I will protest that there are NO geometry problems for which visualization is a detriment. My first thought about the second was that Tri AFC was 1/2 of AEC because AF is the median of AEC. It gives a foot in the door where I know "something". If it leads to a dead end, I still have some information.
Looking at the first problem (square and triangles in inscribed similar shapes, it made me want to pursue the progression for regular n-gon ratios.
Later though, a busy day, Thanks for writing and the nice blog.

Arihant Kothari said...

Nice problem. Worth sharing in mathematics for iitjee section.

CCSSI Mathematics said...

Pat wrote:

>I will protest that there are NO geometry problems for which visualization is a detriment.

We never suggested such, and agree with you that visualization is a key skill--that can be developed through cleverly designed problems, and then usefully applied in myriad situations.

Unfortunately, it is difficult to explain the subtle value of such skills to a student, but it is one of the many possible answers to the perennial "when will we use this?" question.

Pat also wrote:

>And geometry, it seems, often asks us to see a line that isn't there

Unfortunately, not often in American K-12 mathematics education. The New York State Regents geometry course, which is considered comparatively difficult, includes proofs, but the complete diagram is already drawn (can someone point us to a counterexample in a Regents exam 6 point question?)

Common Core's SMP 7 mentions drawing "auxiliary lines", but it remains to be seen how this manifests itself throughout K-12. It's only specifically mentioned once more, in the optional HS (+) standard, G-SRT 9.

You found our soft spot, because we particularly favor problems where you have to draw "auxiliary" lines to aid in solving.

This elementary school problem perplexed many, as a discussion on reddit revealed:

The above-mentioned middle school "fraternal triangles" problem, as well, and continuing to fairly challenging HS geometry proofs:

Pat's Blog said...

CCSSI math, I absolutely LOVE the quadrilateral area problem, and I can see why so many would have a problem with it... Even when they don't have to see what isn't there (it is all there in front of them), they have trouble if it is not where they expect it. Somehow I think if you dotted the given legs of the two right triangles that alone would increase success (and reduce the potential for learning) by 50%...

I admit I had to think hard about the two circles and a triangle.. and imagined a special case to make it all work out.

Thanks for the comment and the links, I will probably return to these when I process a little.