Wednesday, May 2, 2007

Going off on Tangents without Calculus!

[Update: Answers to several of these are now posted in the comments. Also, some nice discussion as well.]

To challenge Geometry, Algebra 2 and Precalculus students, we can always go back to our old friend, coordinate geometry. When I learned this way back when, it was referred to as 'Analytic Geometry'!

The following is a series of problems that review some basics of circle geometry, coordinate methods and lots of good algebra. Most of these can be found elsewhere and there are several different methods of approach. The method I'm suggesting for the first few problems is a bit different, i.e., determining the general equation of a tangent line to a circle, whose center is at the origin, at an arbitrary point on the circle. It used to be a standard formula taught in that above-mentioned course, but few students see it nowadays. Try it as in-depth investigation or exploration, starting in class or as an extension (long-term assignment or extra credit). Our AP calculus students can benefit from 'open-ended' experiences like this before they get to the AP course.


For the first 2 questions, consider the circle whose center is at (0,0) and whose radius is 5.

1. Determine the equations of the tangent lines to this circle at the points (3,4) and (4,3). Write the equations in the form Ax+By = C. What do you notice about the results?
2. Based on the pattern of your answers in question 1, make a conjecture about the equation of the tangent line to this circle at an arbitrary point (x1,y1) on the circle. Now verify your conjecture 'analytically', i.e, using coordinate methods and algebra.

3. Based on the above patterns, make a conjecture about the equation of the tangent line to the circle of radius r, center (0,0) at an arbitrary point (x1,y1) on the circle. Verify your conjecture.

4. Now we return to the original circle of radius 5, center (0,0). Write the equations of the two tangent lines to this circle, which have a slope equal to -2. Again, write them in the form
Note: There are many many approaches here. Discuss at least two!

5. Now, let's go outside the circle. Consider the circle of radius 1, center at (0,0) and let P have coordinates (0,2). Determine the equations of the two tangent lines to the circle through P. Also indicate the coordinates of the points of tangency.
[This 'special' case can be handled with very little algebra or computation.]

6. To generalize a bit more, consider the circle of radius r, center at (0,0) and let P have coordinates (0,2r). Again, determine the equations of the two tangent lines to the circle through P. Also, express the coordinates of the points of tangency in terms of r.

7. Final Generalization: Consider the circle of radius r, center at (0,0) and let P have coordinates (0,b) where b > r. Again, consider the 2 tangent lines to the circle, which contain P. Write an algebraic expression for the coordinates of the 2 points of tangency in terms of r and b.


jonathan said...

Once again, something I am going to lift directly. Thank you.

It is off topic, but vaguely related to work one class did a while ago, and will do again shortly. The problem of generalization (is it a conjecture? have we proved it?) is important, as are some basic facts about circles, and a fresh peek at coordinates.

You do plan to keep posting these after June? (and congratulations)

Eric Jablow said...

Do your students know that perpendicular lines have negative reciprocals for slopes?

I could never stand up and prove geometrically this classic result:

Let P be a point outside a circle C. Let L be any line through P, intersecting C. Let A and B be the two points of intersection. [If L is tangent to the circle, just count the one point of intersection twice.] Then, PA·PB is constant, and does not depend on L.

Dave Marain said...

thanks for the nice words...
By the way, I used the phrase 'verify your conjecture.' I think there is some ambiguity there. Perhaps better to say, 'Justify' or "Prove the validity of your conjecture analytically (algebraically)'?? We know students are confused by these kinds of directions, so what do you use?

Yes, I plan to keep writing these until I get 'writer's blog'! Although it might seem that I will now have more time to think about math problems, my life will be extremely full with 7 children (including 4 teenagers), 4 grandchildren and a few things my wife might want me to get to that I've successfully postponed for 3+ decades while in the classroom! Like cleaning my desk!!

ah, the Power of a Point Theorem - great stuff! Cut-the-knot does a beautiful job of demonstrating this with similar triangles. Yes, students from algebra 1 on know the slope rule for perpendicular lines...

Some answers to the tangents problems:

1. At (3,4): 3x+4y=25
At (4,3): 4x+3y=25

2. (x1)x + (y1)y = 25

3. (x1)x + (y1)y = r^2

4. (2 √ 5)x + (√ 5)y = 25 is one of them; I'll let you figure out the other...
Note: I'm using html unicode here for the radical symbol so it may not appear correct in all browsers. Let me know if you can read the expression as '2 times the square root of 5' etc...

5. I'll provide one of the points of tangency for you:
((√3)/2, 1/2)
Should this look familiar to your 'trig' students!?! One can never escape the 30-60-90 triangle!

Have fun with the others. I would like to see how creative students can be with their approaches here. Never underestimate the 'power of a student' (sorry, Eric, couldn't resist that play on words...)

jonathan said...


your question is an important one. What constitutes proof? I teach my students that proof is "that which convinces" and that the standard might be different in different settings.

However, a conjecture is not proof, and they can tell the difference. (or I hope they can!)

This is a topic we regularly return to (Also, they get it passively, for those few lessons where I do not attempt to let the mathematics convince them, but simply assert rules or facts, because I consciously point out that I am asking them to accept something, without proof)


jonathan said...


it is one of the more annoying results in circle geometry that, given point P and points A and B on a circle such that all three are colinear, that the product of the lengths AP and AB is constant.

Notice that P need not be external to the circle, nor must A and B be distinct.

(although we prove these as three separate theorems. Really we are appealing to the same similar triangles and inscribed angles, but with different looking diagrams)

jonathan said...

I've printed this as a worksheet for later today....

I may provide feedback.

jonathan said...

This worked very nicely! I had these kids in standard groups, gave a stern warning that they had to work actively (no waiting for partners to 'get it'), and then they started.

The writing was fairly clear, and they made their first forays. After a few minutes I called for everyone's attention, stopped them from trying to sketch a tangent and write its equation (this activity was clearly going on in one group, and I am suspicious about a second), and I got one group to articulate the tangent/radius business. Then they all set off, and I mostly ignored them.

Some generalized before they needed to, but it's that kind of class.

There were concerns over notation. Several groups preferred (a,b) to (x1,y1).

In one class period (with a longish homework review) they moved through about half. (I genuinely did not intervene, except the one item I mentioned above). I think we will return to it on Friday.

Dave Marain said...

thank you for trying this out! it will soon be more problematic for me to test my lessons in real time!
I like the tangent problem because it requires a blend of ingenuity and mechanical skill from algebra. I'd be interested in reading some of your students' creative approaches. They're very fortunate to have a teacher who enjoys challenging them and gives the time to explore...

Also, how are my math symbols appearing lately? The html code I'm using may not work on all browsers. I'm only testing it in Firefox on a MAC, so I need input from many readers.

Jackie said...

Aha, I new I'd find something here. Next math team topic: circles & coordinate geometry. I'll be giving the kids the questions on Tuesday. I'll let you know the results.


-- as for symbols, I'm using firefox on a mac and they look fine to me too!