Thursday, April 19, 2007

The 'Power' of Geometry - Ratios of Areas

[Update: The answer, thanks to tc, and an in-depth treatment of this problem now appear in the comments. There are also some thoughts about geometry curriculum and how I develop some of these problems. I would be very interested in reader reactions to this and other problems I have written. Are they of any use for math teachers in the classroom or just curiosities to think about for the moment? Sometimes I feel that many educators just don't have the time in a packed curriculum to be able to give any of these 'enrichment' experiences. I guess I am looking for some validation here to continue writing these...]

A recently released SAT question (for copyright reasons I avoid posting exact SAT questions) motivated me to generalize the result of the problem and provide a challenge for the stronger geometry or algebra student. This problem can be solved several ways, some of which involve some 'messy' algebra. I invite our readers to find a Euclidean method that requires very little algebra and can be done mentally! When giving this type of question to our students, it is natural to want to provide hints when they become frustrated. From my own experience, I've learned to allow them to play around with it for awhile and discuss it in their groups before 'steering' them. An algebraic approach using equations of lines is certainly a worthwhile experience. The 'elegant' method I'm suggesting may not be the most desirable to show them at first. Besides, someone may devise an ingenious approach none of us would imagine if we didn't allow them to explore! Isn't that what teaching really is all about - leading the student to find her/his own path?

Ok, here's the question:
Refer to the above diagram. Lines j and n are perpendicular and contain point P(a,b) in quadrant I. Express the ratio of the area of triangle OPC to the area of triangle OPD in terms of a and b.


Totally_clueless said...

Since the two triangles are similar, the ratio of areas should be equal to the ratio of the squares of the altitudes to the respective hypotenuses= a^2/b^2.

The Euclidean part is in figuring out what the altitudes are.

Or, did I miss something?


Dave Marain said...

you nailed it! I guess you were thinking I wouldn't have posted such an 'obvious superficial' problem!
As I suggested in the post, this was not a difficult problem if one knows their geometry well, as you do, and recognizes basic relationships...
The issue is whether most geometry students would see how 'obvious' this is. I don't think so!

Also, tc, how do you think many geometry teachers feel about the altitude on hypotenuse theorems? Do you think they love teaching it because students find it meaningful, easy to understand and retain for tests, and useful in the real world? Not in my experience!! Outside of a method of proof for the Pythagorean Theorem, I don't believe that many educators enjoy this topic as much as the Pythagorean Theorem itself and the thought that maybe this topic should be deemphasized or even deleted has perhaps flitted across their radar screen. Of course, I am risking inciting the wrath of the pro-Altitudists but that's the danger of writing such a controversial 'inflammathatory' blog! (Sorry, it's around 5 AM and the 2nd cup of coffee hasn't hit me yet...).

A few more points:
1. The original SAT problem gave values for (a,b) like (3,2) and asked for line n. The intent was very different. They were testing basic principles of coordinate geometry and the definition of slope. That question simply triggered my thoughts. That's how I write many of these questions. I see a nice problem and try to generalize the result. Once I get into it, other possibilities or avenues appear and I may go off in a completely different direction. This is I believe what the research mathematician does and it's what I enjoy immensely. How often do our students get the opportunity to explore like this? Do many educators assume it's appropriate for only the most motivated or talents?

The following is a far more complicated algebraic approach, but I believe the intermediate results are interesting. Let me know if you see some deeper meaning in this:

The y-intercept of n is:
The x-intercept of n is:
The area of triangle OPC:
The area of triangle OPD:
The ratio of areas reduces to
(a/b) divided by (b/a) or a^2/b^2.

There's some decent math going on here and I don't think it's time wasted to have students attempt some of these other results outside of class. A decent extension or alternate to the original question. Reviews lots of stuff...
Also, all of these results came from just a single point P(a,b) in the first quadrant! Students should appreciate that.

Totally_clueless said...

Hi Dave,

5am on a Saturday morning! Don't you believe in sleeping in?

Also, my first instinct was to work out the problem in a way similar to what you have elucidated, but in general believe that there are always simpler methods. So, I decided that I would not do the grunge, but rather think some more.

I think most students are happy if they get it correct, rather than necessary looking for elegant methods. That is probably the prerogative of old geezers like me who can do this for fun rather than having the spectre of a grade hanging in the balance.

I think there is a certain beauty to these altitude on hypotenuse problems. You very soon realize that the three triangles are similar, but forget soon after and rederive it over and over. It is pretty neat how you are able to exploit this property to come up with neat problems.



Dave Marain said...

you always know the right thing to say to encourage me to keep writing these problems! if i don't get up at 4:30 to work on these, i will havc little other time during the day!
writing challenging problems that probe for meaning is a labor of love for me but, at the same time, it's 'labor-intensive'. i do hope there are some out there who are considering using these in class to promote reasoning and a more profound understanding of concept...
tc- i agree with your appraisal of the altitude on hypotenuse theorems; more important to me than the specific geometric mean relationships is the notion of similarity...
thanks again for your commments