MathNotations

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ドル.95. Secured pdf will be emailed when purchase is verified.

Posted by Dave Marain at 11:27 AM 0 comments

Labels: algebra 2, circles, Common Core, Desmos, geometry, investigations, tangents

Posted by Dave Marain at 7:21 AM 7 comments

Labels: geometry, tangent circles, tangents

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!

Posted by Dave Marain at 7:18 PM 2 comments

Labels: circles, geometry, math challenge, tangents

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:

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?

Posted by Dave Marain at 7:19 AM 6 comments

Labels: algebra 2, circles, coordinate problems, geometry, investigations, precalculus, tangents

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...
  

Posted by Dave Marain at 7:27 AM 9 comments

Labels: circles, geometric sequence, geometry, infinite geometric series, investigations, recursion, tangents

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?

Posted by Dave Marain at 6:24 AM 4 comments

Labels: circles, geometry, math challenge, tangents

[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.

Posted by Dave Marain at 6:43 AM 3 comments

Labels: 3-4-5 triangles, circles, geometry, investigations, similar triangles, tangents

[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.

Posted by Dave Marain at 5:53 AM 9 comments

Labels: algebra 2, circles, coordinate problems, geometry, investigations, tangents

MAA members will likely recognize the following challenge that appeared on the outside of the envelope in the mailing to members or prospective members. I plan on giving this to my AP Calculus students as review for the exam or afterwards. As usual I will modify it for the student, place it in the context of an activity, broken into several parts with some hints. The original problem comes with a helpful diagram, however, unless I scan it, it would be difficult to reproduce. The problem involves a property of a point on an ellipse and requires basic understanding of the parametric form of this curve and some basic calculus and trig. The last part of the activity suggests a possible significance of this property but I'll leave the details to our astute readers.

Consider a standard ellipse, center at (0,0), with major axis of length 2a on the x-axis and minor axis of length 2b.
Let P(x,y) be a generic point on this ellipse with the restriction that P is not one of the endpoints of the major or minor axes. Consider the tangent and normal lines at P. Let P denote the point of intersection of the normal line with the x-axis and Q, the point of intersection of the tangent line with the x-axis.
Prove that (OP)(OQ) = a²-b², where OP represents the distance between the origin and P and similarly for OQ.
Here is an outline with several parts for the student:
(a) Show that x = acos(t), y = bsin(t), 0<=t<2pi,>2-b²)/a)cos(t).
(f) Use (d) and (e) to derive the desired result: (OP)(OQ) = a²-b²
(g) Explain why we did not allow P to be an endpoint of the major or minor axes.
(h) What does the expression a²-b² have to do with the foci of the ellipse? For EXTRA CREDIT, investigate this 'focal' property further.

Posted by Dave Marain at 8:40 AM 0 comments

Labels: calculus, ellipse, MAA, parametric, tangents, trigonometry

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ドル.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!

Posted by Dave Marain at 6:12 PM 15 comments

Labels: circles, geometry, inscribed, quadrilaterals, SAT-type problems, tangents

Pi day is over, but it seems fitting to continue exploring. Archimedes did more than develop an approximation procedure for pi ! There are many excellent websites that explain the following in greater detail and discuss many more of Archimedes' theorems about parabolas and tangents. I attempted to draw a diagram using Draw in Word. It's crude but you'll get the idea. The object is to share this extraordinary piece of history of mathematics and have your students finish the proof that a light ray from the focus that strikes a parabolic surface is reflected in a ray that is parallel to the axis of the parabola. This is equally interesting in reverse: External light rays and other forms of electromagnetic radiation that are parallel to a parabola's axis are reflected to the focus, very useful for radar and other 'collection' devices.
Considering that Archimedes' proofs used only geometric properties makes his work even more astounding (now of course we can use coordinate geometry, calculus, etc.). This type of investigation is usually deferred to College Geometry courses, but I believe we can deliver it to motivated geometry, 2nd year algebra or precalculus students. If nothing else, it makes for a wonderful long-term project!

Ok, here goes...

In the diagram below, I've gone out of my way to make the reflecting ray NOT look parallel to the axis, even though we're trying to prove it is. This is to help students avoid assuming collinearity, when, in fact, that needs to be proved!

The two angles marked X are equal by a reflection principle (angle of incidence equals...). The two angles marked Y are equal because it can be proved that the tangent line at P is the perpendicular bisector of segment FP', where F is the focus and P' is the foot of the perpendicular from P to the directrix. I chose not to derive Archimedes' very subtle argument, but it is worth studying the proof. The proof starts by constructing the perpendicular bisector and showing that this line passes through P but no other point of the parabola, thus it is tangent. Alex Bogomolny's excellent and in-depth treatment (with java applets) of this topic (on cut-the-knot) is very worthwhile reading.

The student is being asked to prove that the reflecting ray is parallel to the axis. This is equivalent to showing that the line containing PP' and the reflecting ray are one and the same. The argument is straightforward, but students may want to continue learning more about the genius of Archimedes.

[Good luck copying this diagram (jpg). Some of you may find errors in my argument or an extremely simply argument for the parallelism, so pls share!!]

Posted by Dave Marain at 10:02 AM 0 comments

Labels: archimedes, geometry, logic, parabolas, pi, proof, tangents

Posts

Posts

All Comments

All Comments