Desmos Common Core Activity Linking Circles, Tangents and Linear-Quadr Systems

Detailed investigation with extensive background notes for instructor and step by step outline for students to follow. Students will be asked to use a slider to approximate the position of a tangent line of slope -1 to a circle centered at (0,0). The tangent line, x+y=k, requires use of a parameter.

Students will begin with a particular radius, 3, then solve a linear-quadratic system to determine the exact equation of one of the tangent lines. They will also be asked to enter an expression for the other tangent line of slope -1 using the same parameter k. After approximating the locations of similar tangent lines for other radii, they will be asked to solve a general system using radius r.

There are different systems offered to the instructor, depending on the sophistication of the student. Finally, a geometric solution is suggested using 45-45-90 triangles.



 Use new contact form at top of right sidebar to contact me directly!


 If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is $9.95. Secured pdf will be emailed when purchase is verified.

Updates, ODDS AND EVENS and some Geometry Packing Problems

Enjoying your summer hiatus or as busy as ever? I know that feeling!

1. MathNotations' Third Online Math Contest is tentatively scheduled for the week of Oct 12-16, a 5-day window to administer the 45-min contest and email the results. As with the previous contest, it will be FREE, up to two teams from a school may register and the focus for now will be on Geometry, Algebra II and Precalculus. Several other ideas are running through my head but I need the time to bring them to fruition. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' geeeemail dot com."

2. CNNMoney.com Article - Something to tell your students in September!
Here is the link. The 2nd paragraph says it all:

The top 15 highest-earning college degrees all have one thing in common -- math skills.

3. Silly Instruments for Math Teachers to Play
I always told my students that I'm predominantly left-brained -- analytical, organized, detailed, process-oriented, algebraic -- as opposed to most of my children and my wife who are creative, spatial, mechanical, who see the forest more than the trees. One of my sons is a musician and another is a dancer so we are not always on the same wavelength! So I mentioned to my SAT students that I wish I had a more creative side and perhaps be able to play an instrument, but, in fact, the only thing I can "play" is my iPod! One of my students in the front row immediately responded, "I know an instrument you can play, Mr. M -- the triangle! I congratulated her for the cleverness and told her that maybe I will learn how to play the "cymbals." (the class actually applauded that lame attempt at word play!). In fact, I've read that many famous mathematicians were also musicians, so let us know: Do you play an instrument or are passionate about music or do you have a silly instrument for a mathematician to play?

4. Circle Packing Problems
Even though I am dominantly left-brained, I still enjoy challenging spatial geometry problems. I find these questions have improved my creativity and my spatial sense and they often involve multi-faceted thinking. Here are a couple of famous 'packing' problems which are accessible to geometry students. More important than solving these is to give our students a sense of the importance of packing problems and the ongoing research in this area. There are still unsolved problems here!

Although you can easily research packing problems on MathWorld and Wikipedia, the diagrams below come from an exceptional website I discovered. The author, Peter Szabo (missing accents), provides diagrams for packing 2-100 circles with accompanying data (radii, density, etc).


PROBLEM I


The two congruent circles at the left are actually enclosed in a unit square which is not shown.The circles are tangent to each other and to the sides of the square. If these circles have the maximum radius possible, determine the radius.
Note: The indicated square (assume it is a square) is helpful in solving the problem. Trig is not necessary here.

Answer (Yes, I'm providing this since the objective is to discuss the method):
[The following is the diameter, not the radius, of each circle. Thanks to watchmath for correcting this error].









PROBLEM II



Again, imagine that the three congruent circles at the left are enclosed in a unit square and are tangent to each other and to the sides of the square. If the circles have the maximum radius possible, determine this radius.
Notes: The indicated square again may be helpful to solve this problem. Trig can be used but clever use of special right triangles is preferred.

Answer:
[The diameter is given below, not the radius. Thanks to watchmatch for correcting this]

"On The Road Again" With 'TC' -- A Real World Application of Geometry

As my devoted readers know, Totally Clueless, affectionately known as TC, has contributed many insightful comments and profound ideas for us to think about. His sobriquet belies a brilliant creative mind of course. He recently sent me a geometry problem which was motivated by his own experiences driving to work. The problem itself is accessible to advanced middle and secondary students but the result is interesting in its own right and should generate rich discussion in class. I recommend giving this as a group activity, allowing about 15 minutes for students to work on, then another 15 minutes to discuss it. Save it for an end-of-year activity or bookmark it for the future. Beyond the problem, there are important pedagogical issues here:

• How to introduce this
• Asking questions to provoke deeper thought
• Drawing conclusions and further generalizations
• Connecting this problem to other circle or geometry problems
  
• Maximizing student involvement
  

I told TC I would need some time to rework the original problem for the younger students so here goes...





Diagram for Parts I and II







Part I (middle and secondary students)
In my city, there are two circular roads "around the center" of the city, of radii 6 and 4. There are a number of radial roads that connect the two loops. Points A and B in the diagram above are at opposite ends of a diameter of the outer loop and the dashed segment is a diameter of the inner loop.

If I have to go from point A to point B on
the outer loop, I have two options:
(1) Drive along the outer loop (black arc in diagram) OR
(2) Drive radially (blue) to the inner loop, drive along the inner loop (red), and then drive radially out (blue). (Assume that there are radial roads that end at point A and point B).

Show that Option 2 is shorter than Option 1.

Part II (middle and secondary students)
Same diagram but now the radii are R and r with R > r.
Show algebraically that Option 2 is shorter.


Part III (secondary students)












To generalize even further, points A and B are distinct arbitrary points on the circle, central angle AOB has radian measure θ where θ ≤ π. OC and OD are radii of the inner loop; OA and OB are radii of the outer loop. Again the radii of the two circles are R and r, where R > r.

As before, there are two options in going from A to B:
(1) Drive along the outer loop (black arc in diagram) OR
(2) Drive radially from A to C (blue), then along the inner loop from C to D (red), then radially outward from D to B (blue).

Show that Option 2 will be shorter provided π ≥ θ > 2.

Click Read More for further discussion...


Further Comments
(1) TC's original problem was Part III. I decided to add Parts I and II to provide 'scaffolding' for students. Was this really necessary in your opinion?
(2) The results of these questions are independent of the actual radii. TC felt this was an interesting aspect of this problem and I agree. Do you think students will be surprised by this? Do we need to point this out to them? Are there other circle problems you can recall which have a similar feature?

Thanks TC for providing us with another stimulating challenge!

...Read more

SAT/PSAT Geometry Practice: Circles and Similarity

With the PSAT rapidly approaching, here are a couple of problems which require the student to review their knowledge of circles. This will be set up as an open-ended investigation with several parts, but the content is often assessed on standardized tests. As usual there are many approaches, although efficient use of ratios and proportions is the goal here. It is critical that students thoroughly read the detailed given info in the text box.







Part I
In Figure I, determine the length of minor arc PQ.

Part II
In Figure II, determine the area of sector OPQ.

Comments:
A central theme here is the relationships among the ratio of the radii of the two circles, the ratio of their intercepted arc lengths and the ratio of the areas of their corresponding sectors. Students need to have a clear understanding that one is a linear relationship and the other is a direct square variation. there are many ways to set up the solutions of these problems. We will discuss this further in the reader comments.

Also, a good discussion point is to have students explain why the last piece of given information, regarding the central angles not being congruent, was not necessary. It might be interesting to have students compute the degree or radian measures of the central angles.

Reviewing Geometry for Class or SATs - Just a little tangent exercise?

The following problem is certainly appropriate for later in the year when geometry students reach this topic but it can also be used to review a considerable number of essential ideas in preparation for SATs, ACTs or just review in general. It's at the top end on the difficulty scale for these tests, but it's far from the AMC Contest!


Clarifications: Figures are not drawn to scale and the measure of ∠TAU is given in each diagram.

For each of the figures above, determine the following:
(a) the radius of each circle
(b) the length of minor arc TU in each circle

Have fun discovering a variety of approaches!

Variations? Generalizations? Choosing an angle other than special cases like 60 or 90 generally requires trig -- not that there's anything wrong with that!

Setting the Tone in Precalculus - Another Coordinate Investigation

Note: Read the first comment I posted which suggests a purely Euclidean geometry approach to this problem...

Don't forget our MathAnagram for Aug-Sept. Thus far we have received a couple of correct responses. You are encouraged to make a conjecture!
Look here for directions. Here is the anagram again:

PRINCE? NAH! E-ROI!

Tangent problems are usually the domain of calculus but we can keep them within the reach of geometry and algebra if we restrict our attention to circles. The calculus student spends a considerable amount of time solving a wide variety of "tangent to the curve" exercises. As any calculus instructor will tell you, many of the harder problems ask students to determine the equations of the tangent to a curve from a point not on the curve. The issue there is not the calculus. It's all about an understanding of the interface between the algebra and geometry, the essence of coordinate methods. I developed this investigation specifically to address this issue before students enter calculus. Might be another "fun" problem to start the year off with. If nothing else, it will establish the rigor of your precalculus course early on!

Part I of Investigation
Determine the coordinates of the points of tangency for the tangent lines to the unit circle from the point (0,2).
Note: Unit circle refers to the circle of radius 1, center (0,0).

The remaining parts will all refer to this same circle.

Part II of Investigation
Repeat part I for (0,3) and (0,4).
Write your observations, conjectures.

Part III of Investigation
Show that the y-coordinate of the points of tangency for the tangent lines to the unit circle from the point (0,k) is 1/k, where k ≥ 1.

Notes, Comments...
(1) The result of Part III suggests that as k increases, the y-coordinate of the point of tangency decreases (inverse ratio). Ask students what happens as k approaches 1.
Students should make sense of this visually by sketching tangent lines from various points on the y-axis above the circle.
(2) There are several effective methods for solving the above parts, however, one needs to know the fundamental relationship between a tangent line and the radius drawn to the point of tangency. From that point on, one can represent the slope in two ways or represent the y-coordinate of the point of tangent in two ways. This requires strong understanding of coordinates, graphs and algebraic relationships. You may find other methods -- share them! BTW, one could also use trig methods.
(3) I chose the unit circle and a point on the y-axis for simplicity so that the student could focus on essential ideas. However, one could generalize the result to any circle and any point outside. Have fun with that!
(4) Anyone mildly surprised by the reciprocal relationship between the y-coordinate of the point on the y-axis and the y-coordinate of the point of tangency? Can anyone make sense of that?

Squeezing Circles Into the Corner: An Infinite Sequence Investigation in Geometry

Another summer diversion from geometry...

The number of variations for tangent circles is endless and this is one of my all-time favorites. Math contests and SATs seem to have a preference for circles inscribed in squares or tangent circle problems and this one is along those lines. However, the real payoff comes from developing recursive thinking leading to an infinite geometric sequence and its sum! Students will be asked to intuitively "guess" the value of this infinite sum and to then verify their conjecture. Proving it requires nothing more than the classic formula for the sum of an infinite geometric series but, at the outset, this problem is eminently suitable for your geometry classes. Don't hesitate to use it in your "regular" classes. Questions that are deemed appropriate only for honors classes are often suitable for most students if the groundwork is laid (background, examples, etc.) and hints are given strategically.

PART I In the diagram above the larger circle has radius 1, the two circles are tangent to each other and to the two perpendicular segments (you can think of the larger circle being inscribed in a square if you wish).

(a) Make a conjecture from the diagram without computing: The ratio of the radius of the smaller circle to the larger is approximately
(A) 0.05 (B) 0.15 (C) 0.25 (D) 0.35 (E) 0.5
Note: This part may be omitted.

(b) Show that the radius of the smaller circle is exactly (√2 - 1)² = 3 - 2√2
How was your conjecture?
Note: Your decision about giving them the result like this. Obviously if they see part (b) on a worksheet, their estimate in part (a) will be pretty good! My intent was to focus on the method. Of course, feel free to rephrase this.

PART II
Of course we will not stop at 2 circles! Squeeze a third circle into the corner between the 2nd circle and the right angle. Determine its radius by using the result from part (a). [The key here is to think ratios!]

PART III
If we label the radius of the largest circle R₁, the radius of the 2nd circle R₂, the radius of the 3rd circle R₃, etc., we can now define an infinite sequence of these radii.
(a) Find a formula for the nth term of this sequence, n = 1,2,3,..

(b) What is the mathematical terminology for this type of sequence?

(c) Think intuitively here: From the diagram, what should be the "sum" of the original radius R₁ = 1 and the diameters of the remaining infinite collection of circles. [Another formulation: As n-->∞, this sum approaches what number?]

(d) Using the formula for the sum of an infinite geometric series, verify your conjecture in (c).

Comments:

• As always, feel free to use this with your students and revise as you see fit. However, pls use the attribution in the Creative Commons License as indicated in the sidebar.
• Finding the radius of the 2nd circle is a challenge by itself and the problem could stop there. The extensions can be assigned as a long-term project or for those wishing to do extra credit. I always liked having additional challenges for the students who were capable of going further, although relating this problem to geometric sequences or series is of importance. Of course, I am well aware of time constraints faced by the instructor.
• Your thoughts...
  

Figure Not Drawn To Scale! An SAT-Type Geometry/Summer Diversion

In the circle at the left, O is the center, A, B and C are on the circle and OABC is a parallelogram. If AB = 6, what is the length of segment AC (not drawn)?

(A) 3√2 (B) 3√3 (C) 6 (D) 6√2 (E) 6√3




POINTS TO PONDER

Is this an appropriate standardized test question?

Are you an opponent of multiple choice (aka, "multiple guess") questions. Why?

We can also say much about the issue of drawing figures that do not appear to be what they are? Is it just the testmaker's way of misleading or trapping students or is there a valid purpose to this?

Your thoughts about this problem...

A Geometry Classic - Chord and Tangent Riddle

Don't forget to submit your solution to this month's Mystery Mathematicianagram (ok, so I can't decide on a name yet!). We've received 3 correct solutions thus far and I will announce winners around the 20th.







As we wind down the school year, the problems below may come too late for students taking their final exams in geometry, but you may want to hold onto this classic puzzler for next year. I don't consider these overly challenging but I do feel they demonstrate some important mathematical ideas and problem-solving techniques. Further, encourage students to justify their reasoning since some may make assumptions from the diagram without verification. This will review some nice ideas from circles.

OVERVIEW OF PROBLEMS (see diagram)
For both questions, assume the circles are concentric, segment PQ is a chord in the larger circle and tangent to the smaller.

PART I
If PQ = 10, show the difference between the areas of the 2 circles is 25π.

PART II (the converse)
If the difference between the areas of the circles is 25π, show that the length of PQ must be 10.

Notes
(1) It is important for students to recognize that there are many possible pairs of concentric circles (varying radii) satisfying the hypotheses of these problems, yet the conclusions are unique! Some students will assume a 5-12-13 triangle is formed (not a bad problem-solving strategy), but stress that this is not the only possibility! Remember, we're not restricting the radii to integer values.

(2) There is a classic math contest strategy for these questions that mathematicians love to employ - the "limiting case." Can you guess what I mean by this phrase?

Geometric "Connections" - How one problem leads to another...

Answers to Problems from Previous Post:

(a) Substitute 0 for x, -1 for y in both equations.

(b) a > 1/2

(c) Two points above x-axis: a > 1; Below x-axis: 1 > a > 1/2; On x-axis: a = 1
Note: If 1/2 > a > 0, then the only point of intersection would be (0,-1).

(d) x = ±[√(2a-1)]/a; y = (a-1)/a

(e) Points: (±4/5 , 3/5); a = 5/2
Note: Pls check for accuracy!

Now for the connection...


In the previous post, we were given that the radius of the circle was 1 and the area of the triangle was 32/25. From this it can be shown that PQ = 8/5 and, in fact, the quadrilateral PQRS shown in the figure at the left is a square whose area is 64/25. The fact that this was a square intrigued me. I hypothesized that, up to similarity, these numbers were unique. This led me to the diagram at the left and the following converse of the previous problem.
Note: This problem is now unrelated to the parabola.

In the diagram above, points P and Q are on the circle, PQRS is a square and segment SR is tangent to the circle at T.
If the radius of the circle is r, show that the area of the square is (64/25)r², and, consequently, the area of ΔPQT = (32/25)r².

Comments:
(1) Students should not find this overly challenging using standard methods for solving circle problems (and the fact that it is closely related to the previous question).
(2) Of course, what really intrigued me is how, once again, the 3-4-5 triangle recurs! Ask your students to find a triangle in the diagram similar to 3-4-5. They need to draw something but this should occur naturally from the standard solution to the problem.

Circles, Chords, Tangents, Similar Triangles and that Ubiquitous 3-4-5 Triangle

[As always, don't forget to give proper attribution when using the following in the classroom or elsewhere as indicated in the sidebar]

The cone in the sphere problem led me to an interesting relationship in the corresponding 2-dimensional case with a surprise ending. (Only a math person would compare a math problem to a mystery novel!). The following investigation allows the student to explore a myriad of possibilities: from similar triangles to the altitude on hypotenuse theorems to Pythagorean, to chord-chord or secant-tangent power theorems, coordinate methods, draw the radius technique, etc. Sounds like this one problem might review over 50% of a geometry course? You decide for yourself! Just remember -- one person is not likely to think of every method. Open this up to student discovery and watch miracles unfold...

STUDENT ACTIVITY OR READER CHALLENGE
In the diagram above, segment AF is a diameter of the circle whose center is O, BC is a tangent segment (F is the point of tangency), BC = AF and BF = FC. Segments AB and AC intersect the circle at D and E, respectively. Lots of given there! Perhaps some unnecessary information?

(a) If AF = 40, show that DE = 32.
Notes: To encourage depth of reasoning, consider requiring teams of students to find at least two methods.

(b) Let's generalize (of course!). This time no numerical values are given. Everything else is the same. Prove, in general, that DE/BC = 4/5.

(c) So where's the 3-4-5 triangle (one similar to it, that is)? Find it and prove that it is indeed similar to a 3-4-5.

The Power of Visualization: Another View of Inscribed vs. Circumscribed

A student, Melissa, in my SAT class, shared her way of thinking about a well-known problem that appeared on this blog awhile back as part of a larger investigation. You may recall the post about the ratio of the areas of circumscribed and inscribed figures.

I told her I would celebrate her method on my blog, which would definitely be viewed by less than or equal to a million people a day! One of my students asked me why I don't simply write a book about all these ideas and I replied, "I'm thinking about it." He asked what the title would be and I replied, "What I've Learned From My Students."

Here's the problem/investigation/challenge/activity///// for you or your geometry students:

Consider the diagram above. Assume that it depicts two squares. The smaller inscribed square is formed by joining the midpoints of the larger square.

(1) Explain why the area of the inscribed square is one-half of the area of the larger.
Easy so far...

(2) Consider the circle circumscribed about the smaller square, i.e., it passes through its 4 vertices. Explain why this circle is inscribed in the larger square. This requires that you show that the 4 sides of the larger square are tangent to the circle. [By the way, most students would assume this is obvious from the diagram, but ...]

Not impressed? Deja vu all over again as Yogi would say? This is how Melissa demonstrated that, for a given circle, the area of the circumscribed square is twice the area of the inscribed square. Is this the method you have seen or used yourself? Consider that she used no variables, didn't plug in particular values for the dimensions, etc. She drew the diagram and basically said that the diagram proves itself! A proof without words, so to speak (that might even make Sidney Kung proud). I congratulated this young lady and the class applauded. Interesting how students immediately recognize someone's brilliance...

Inside or Outside? A Geometry Investigation...

[A special thanks to Mike for correcting the error I made in question 3 below. It has now been updated!]

BACKGROUND
Students in geometry often see problems involving inscribed and circumscribed circles, squares, triangles and other polygons. Questions involving such diagrams often appear on standardized tests and on math contests as well. Although this particular post focuses only on ratios of areas involving squares, equilateral triangles and circles, I am planning a series of investigations which delves far more deeply, requiring students to discover and verify more general relationships for polygons of n sides. Further, as the number of sides of the polygons increase, students will be asked to analyze the ratio of the areas of the circumscribed to the inscribed polygons and consider if they approach a limiting value. Thus this series of activities prepares students for the calculus as well. Students are also introduced to the duality principle, useful for later study in advanced geometry. A strong background in geometry is needed and, at some point, a knowledge of trigonometric ratios is required. This could be a culminating project for a marking period or the year. I hope you enjoy this and will save it for the school year or ...

Note: Although it would appear to be more logical to have students determine ratios for the triangles first (n = 3), the diagram shows the squares (n=4) on top because the analysis is somewhat easier.

Note: Although this is couched as an investigation for the classroom, my readers are invited to attempt the questions below and suggest various approaches to finding the ratios. Those experienced in this topic will find each question fairly straightforward, however, consider the bigger picture here! Those whose geometry is rusty will need to review some basic properties of circles and polygons.



FOR THE STUDENT:
1. The diagram at the upper left depicts a circle circumscribed about a square and a second circle inscribed in the square. Verify that the ratio of the area of the inscribed circle to the area of the circumscribed circle is 1:2.
2. The diagram at the upper right may be thought of as the dual to the first diagram, in that each circle has been replaced by a square and the square by a circle. Show that the ratio of the area of the inscribed figure to the area of the circumscribed figure is invariant, that is, it remains 1:2.
3. The diagram at the lower left depicts a circle circumscribed about an equilateral triangle and a second circle inscribed in the triangle. Show that the ratio of the area of the inscribed circle to the area of the circumscribed circle is 1:4.
4. Similar to #2 except that the circles and the triangle have now been interchanged. Again, show that the ratio is invariant.

When Curves Collide: Quadratic Systems Explored...

[Another Update: Mutiple solutions to (a) and (b) are now provided in the comments (as of 10:50 PM 5-17-07).]

Target Audience: Algebra 2, Advanced Algebra, and beyond...

The previous post challenged students to consider 'basic' properties of circles and triangles. Now we will look at systems of quadratics - circles and parabolas in particular. The purpose here is to help students go beyond the standard algorithms of solving systems, by analyzing a general type of system using a parameter r. The use of parameters has become the norm on the AP Calculus Exam. Algebra students may benefit from an early introduction.


OVERVIEW FOR INSTRUCTOR
Consider the system:
x² + y² = r²
y = r² - x²

Here, r denotes a positive constant. Depending on the value of r, this system will have either 2, 3, or 4 solutions!

Simple problem of a parabola intersecting a circle, right? We will assume here that students have already solved specific cases of such systems both graphically and algebraically (substitution, etc.). They have been shown that parabolas and circles may intersect in 0, 1, 2, 3, or 4 points. As a review, begin this investigation by asking pairs of students to sketch (no equations here) graphs depicting each of these cases. That visualization is a powerful context for the algebraic solutions and may motivate the students to consider why varying the parameter r in this lesson leads to different conclusions. This type of analysis goes beyond standard problems and prepares students for the open-ended free-response types of standardized or AP questions they will later encounter in high school or college.

FOR THE STUDENT:

(a) r=1
Solve the following system first graphically, then algebraically:
x² + y² = 1
y = 1 - x²
This system demonstrates that a quadratic-quadratic system may have ________ (number) solutions.
For this system, the points of intersection are ___________________________.
For x between -1 and 1, the parabola is (above, below) the circle.

(b) r=2
Solve the following system first graphically, then algebraically:

x² + y² = 4
y = 4 - x²

This system demonstrates that a quadratic-quadratic system may have ________ (number) solutions.
For this system, the points of intersection are ___________________________
Restrict the domain to -2 ≤ x ≤ 2. For what values of x is the parabola above the circle in this system? Below the circle?

(c) What set of positive values of r have we not yet considered?
For such values of r, make a conjecture about the number of points of intersection of the system:
x² + y² = r²
y = r² - x²

Now choose a particular value of r in this set, say r = 1/2. Check the validity of your conjecture by solving the system for this particular value both graphically and analytically (algebraically).

(d) Analyze, algebraically, the following system for all positive values of r. Show carefully that your algebraic solution leads to 3 distinct cases for r:
x² + y² = r²
y = r² - x²
For each case, give the solutions (ordered pairs) in terms of the parameter r.
Explain why, for all positive values of r, there will always be at least TWO solutions of this system. That is, the possibility of zero or one solution does not exist for this system...

A Simple Geometry Proof or Circular Reasoning??

[Update: Partial solution now in the comments. Fascinating discussion taking place about this innocent-looking problem...]

Figure 1

A standardized test sample problem I saw this evening, got me to thinking, which is often very dangerous. I will pose this 'open-ended' problem in terms of Jake making an assertion and Jack trying to convince him he's wrong. Jake keeps arguing and so does Jack. Who will win the argument logically?

Jake shows Jack a piece of wood he cut out in the machine shop in the shape of a circular arc bounded by a chord (See Figure 1 above). Jake claimed that the arc was not a semicircle, and, in fact, he claimed it was shorter than a semicircle, i.e., segment AB was not a diameter and arc ACB was less than 180 degrees. Jack knew this was impossible and argued: "Don't you see, Jake, that O must be the center of the circle and that OA, OB and OC are radii!" Jake wasn't buying this since he measured everything precisely. He argued that just because they could be radii didn't prove they had to be!

Here's your challenge for today:

(a) Find at least THREE different ways to PROVE that Jake is wrong, i.e., AB, in Figure 1, must be a diameter and O is the center. [Note: Can one assume perpendicularity?]

Figure 2

(b) Assume, in Figure 2 above, that PQ is a diameter, O is the center, chord AB is parallel to PQ and radius OC ⊥ AB. Determine the length of segment CD, and in particular, show that CD < DB.

(c) Refer to Figure 2, but let's generalize by removing the numerical values. Prove, in general, that for any chord AB parallel to and above diameter PQ, CD < DB.
[Pls note: We are no longer requiring that the length of chord AB be half of the diameter!]

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.

STUDENT ACTIVITY

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
Ax+By=C.
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.

Challenging Geometry: Circles Inscribed in Quadrilaterals, Right Triangles

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar.  175 problems divided into 35 quizzes with answers at back. Suitable for SAT/Math Contest/Math I/II Subject Tests and Daily/Weekly Problems of the Day. Includes both multiple choice and constructed response items.
Price is $9.95. Secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL (dmarain "at gmail dot com") FIRST SO THAT I CAN SEND THE ATTACHMENT!

------------------------------------------------------------


Update²: See the awesome article in MathWorld on tangential quadrilaterals for more info re problem #2.
Update: See Comments section for some answers, solutions.

Part (a) of each of the following are somewhat difficult questions that can be found in some geometry textbooks. These are numerical exercises and good practice for the more difficult SAT-types of questions or for math contests. The last part of each question is an extension or generalization of the problem. Texts do not often ask students to delve beneath the surface and look for general relationships.

1. A circle of radius 4 is inscribed in a right triangle with hypotenuse 20.
(a) Find the perimeter of the triangle without using the Pythagorean Theorem. Justify your reasoning.
(b) Using the Pythagorean Theorem, show that the triangle is similar to a 3-4-5 triangle.
Note: Many students tend to guess multiples of 3-4-5 when doing these. Sometimes they get lucky but they need to prove it!(c) PROVE in general that the perimeter of a right triangle is twice the sum of its hypotenuse and the radius of its inscribed circle. Again, no Pythagorean Thm allowed.
Note: There are well-established formulas for the inradius of a triangle. Our objective here is to look at one special case.
2. A circle is inscribed in a quadrilateral which has a pair of opposite sides equal to 12 and 18. Neither pair of opposite sides of the quadrilateral is parallel.
(a) Find the perimeter of the quadrilateral. Justify your reasoning.
(b) PROVE in general that the perimeter of a quadrilateral in which a circle is inscribed equals twice the sum of either pair of opposite sides.
Note:: Not all quadrilaterals have an inscribed circle, so this is a strong condition.

Note: As always, these results need independent verification. I welcome your comments and edits!

Lattice Points Problem Part 2: Circles, Gauss' Circle Problem and Pick's Theorem

Eric Jablow inspired me to develop the following extended enrichment actvity/project for Geometry students. You can research the general solution of Gauss' Circle Problem in MathWorld but for this application it isn't necessary to use that formula.

Consider the circle of radius 5 centered at the origin.

(a) Determine the coordinates of the 12 lattice points on this circle. Recall that lattice points are points whose coordinates are both integers.
Note: Is it reasonable from symmetry arguments that the number of lattice points is divisible by 4?
(b) Using graph paper, show there are 69 lattice points INSIDE this circle. Describe your counting method.

The total number of lattice points inside or on our circle is 69+12 = 81. This agrees with the result from Gauss' formula but we will now 'approximate' this result using Pick's Theorem which gives the relationship among interior, boundary points and the area of a polygon whose vertices are lattice points.

(c) Consider the inscribed dodecagon formed by connecting the 12 lattice points from part (a). Determine the lengths of the sides of this polygon.

(d) By dividing the polygon into 12 triangles show that the area of this polygon is 74. No trigonometry, just Pythagorean and basics!
Hint: This is not a regular polygon but you can still divide it into 12 isosceles triangles.
Comment: Do you find it surprising that the area is rational (in fact, integral), considering that the sides are irrational?

(e) Pick's Theorem states that the area of a polygon whose vertices are lattice points is given by the formula A = I + B/2 - 1 where I = the number of interior lattice points and B = the number of boundary lattice points, that is, points on the polygon.
Show that Pick's Theorem leads to I = 69.

(f) For our problem the number of lattice points inside the circle matched the number of points inside the inscribed polygon. A coincidence? Whether it's true or not, explain why this result seems to make sense.

(g) To further investigate this 'coincidence', change the radius to 10.
(i) Show, by counting, that there are 317 lattice points inside or on this circle.
(ii) Show that there are still 12 lattice points on this larger circle.
(iii) Show that there are 303 lattice points inside or on the resulting dodecagon.
[As before, find the area w/o trig and use Pick's Thm. Note: To find the area of the polygon use (d) and ratios!]
(iv) Show that the point (4,9) is inside this circle but outside the dodecagon! This suggests why Pick's theorem fails in this case! Why?

Good luck! This is an extended challenge that I will leave up for several days and invite comment. Whether you implement it in a classroom or not, enjoy!

[By the way, some of the numerical results (like 303) have not been independently verified. If you find an error, let me know!]

Challenge Problem for 2-7-07

Day 8 - still awaiting a response from the National Math Panel...

Pls read the comments re the problems from 2-6-07. There are hints for #1 and a good discussion about #3.

Something different today. This is another one of those weekly online challenges I gave last year just before the AMC Contest. Students found it difficult but those who persisted got it. Perhaps that's the best reason to give these challenges -- to teach persistence, a quality that distinguishes some of the best researchers and problem solvers from the rest. Remember to click on the image to magnify it if it's too small.