This blog concerns how to determine the intersection between two circles in the plane algebraically. It is a problem that can crop up in a variety of situations, from gaming to tools for computer aided design to astronomy.
This problem is interesting because is it a conceptually simple problem whose algebraic formulation is nonetheless apparently complex: a system of non-linear equations that are quadratic in both variables. However, by doing some geometric analysis of the problem, and applying tools from vector geometry, we are lead to a specific mathematical transformation of the problem that radically simplifies it. The key idea turns out to be a specific change of basis.
My exposition of this uses a few diagrams (which are not supported by the blog editor) and a lot of mathematical expressions (which are clumsy to create in the blog editor), so I put it in an Adobe Portable Document Format (PDF) document that you can access using the following...
Proof of the Assertion that Any Three Non-Collinear Points Determine Exactly One Circle
This is an interesting problem in geometry, for a couple of reasons. First, you can apply some earlier, basic geometry principles; and secondly, you can choose two different strategies for solving the problem.
The basic geometry underlying: any three non-collinear points determine a plane, somewhere in 3D space. Once that has been done, imagine that the plane has been rotated into the x-y plane, which will make the problem much easier to solve!
The two strategies for solution are: (Proof A) actually solve to find the circle. This is equivalent to finding the center of the circle (finding the equation of the circle is simple from there). But, you actually have to do some math to get this! If, while doing this, there is no possibility to obtain other values for the coordinates of the center of the circle, you have proved the assertion as well as obtained a method (and perhaps...
This is another way to find a distance between two parallel lines. This derivation was suggested to me by Andre and I highly recommend him and his answers to any student, who wants to learn math ans physics. This derivation requires the knowledge of trigonometry and some simple trigonometric identities, so this may be suitable for more advanced students.
Once again, we have two lines.
y=mx+b1 (1)--equation for the first line.
y=mx+b2 (2)--equation for the second line.
Now recall that the slope of the line is the tangent of an angle this line forms with the x-axis. Indeed, m=(y2-y1)/(x2-x1), where x1, x2, y1, y2 are the x- and y-coordinates of any two distinct points on the line. If one draws the picture, it will be immediately obvious that m is the tangent of the angle between the line and the x-axis.
The difference b2-b1 gives the relative displacement along the y-axis of two lines....
Suppose, one have two parallel lines given by the equations:
y=mx+b1 and y=mx+b2. Remember, if the lines are parallel, their slopes must be the same, so
m is the same for two lines, hence no subscript for m. How would one approach the problem of finding the distance between those lines?
First, if one draws a picture, he or she shall immediately realize that if a point is A chosen on one of the lines, with coordinates (x1, y1), and a perpendicular line is drawn from that point to the second line, the length of the segment of this new line between two parallel lines give us the sought distance. Let us denote the point of intersection of our perpendicular line with the second line as B(x2,y2).
What do we know of point A and B?
First, since A lies on the first parallel line, its coordinates must satisfy the equation for the first line, that is,
In the spirit of giving, starting on 11/29/2013, I will be offering a few brainteasers/ trivia questions where the first 3 people to email me the correct answer will receive a free, one hour, tutoring session in any subject that I offer tutoring for (via the online platform)! That's right free! Get your thinking hats on everyone!
Andrew L. Profile
Area, Volume and Circumference equations:
Area of a Square
Area of a Triangle
Area of a Rectangle
Right Triangle/Pythagorean Theorem
Area of Parallelogram
Area of a Trapezoid
Area of a Circle
Circumference of a Circle
c=πd or c=2πr
Volume of a Sphere
Surface Area of a Sphere
Volume of a Cube
Volume of a Rectangular Solid
Slope of a line Equations
m is the slope
b is the y-intercept
y is a y coordinate on the graph (that coincides with the line)
x is an x coordinate on the graph (that coincides with the line)
I've heard this sentiment over and over--sometimes from students, and sometimes, I'll admit, in my own head.
Last night, I was working on my own math homework, and there was one problem I just couldn't get my head around. I read the book, looked back at my class notes, and even sat down with a tutor for a while, and still, when I tried a new problem of the same type on my own, it just didn't work!
"Maybe I'm not as good at math as I thought," I told myself. "Am I REALLY smart enough for bioengineering?"
It was hard, but I told myself "YES!" And I kept working. I laid the assigned problems aside and started doing other problems of the same type from the book. I checked my work every time. Each problem took at least ten minutes to solve, and the first three were ALL wrong! I kept going. I got one right, and it made sense! I did another, and it was half right, but there was still a problem. I did another, and it was right!
I graduated with a BS degree in Chemistry from East Carolina University, Greenville NC. Since then, I worked in pharmaceuticals as a chemist/lab analyst for 1.5 years performing drug analyzes with High Performance Liquid Chromatography, Ultra-Violet Spectrometry, FT-NMR IR, and moisture testing. Also I tutored General Chemistry I and II /Math(pre-algebra, algebra I & II, calculus, geometry at a community college. Also I was a General Chemistry II Lab Instructor at the college and have taken Human Gross Anatomy. If anyone of you need help with Anatomy, let me know! I am a more a visual learner and use concept mapping frequently. I thoroughly enjoy helping students achieve their learning and for them to be able to reach their potential. I look forward in working with you.
Most recently, I was working with a student on the angles in a triangle. We started with a circle to show the angles all the way around, their sizes, and how they compare to each other. Then, I set my arms hand to elbow in an L shape. We worked on showing the sizes of various angles by moving my hand at 12-o'clock (90 degrees) to a position that would show the angle size when my other arm was at 9-o'clock (zero degrees). It worked very well.
As the school year ramps up again, I wanted to put out a modified version of a Memo of Understanding
http://en.wikipedia.org/wiki/Memo_of_understanding for parents and students. It seems each year in the rush to get through the first weeks of school parents and students forget the basic first good steps and then the spiral downwards occurs and then the need for obtaining a tutor and then the ‘wish for promises’ from a tutor. Pay attention to your child’s folder or agenda book. A student is generally not able to self regulate until well into high school. Some people never quite figure it out. Be the best person you can be by helping your child check for due dates, completeness, work turned in on time. Not only will this help your child learn to create and regulate a schedule, it prevents the following types of conversations I always disliked as a teacher ("Can you just give my child one big assignment to make up for the D/F so they can pass"; "I am going to talk to...
When given two ratios (in the form x:y) or two relations (in the form of fractions), if the ratios of each element are the same they're said to be proportionate.
Example: 3/6 and 1/2 are proportionate because 3 out 6 is the same as 1 out of two (half).
When given two fractions to prove as proportionate, such as
you solve through cross-multiplication.
Cross multiplication involves multiplying the numerator (number on top) by the denominator (number on bottom) of the other fraction, and then comparing the results. If the values are the same, the fractions are proportionate.
The set-up above will be set-up as such:
1 * 6
2 * 3
Because both values are the same, these fractions are proportionate.
I was a fairly typical young person and, like my peers, counted down the days until summer. My mother was a math professor, so I never stopped doing math during the summer, but felt like other parts of my brain became a little mushy in the summer. Come September, it was difficult to get back into the swing of writing papers and studying history and memorizing diagrams. I was out of practice and lost my routine. As an adult, I have almost continually taken classes, because I enjoy learning and find that from class to class, I need to maintain a routine, i.e. a study area and a time of day that I complete my assignments. I have also found that reviewing material a week or two before the course begins helps me to start the class with more confidence and competence. I am a big believer in confidence fueling success and I wonder if younger students practiced assignments in the week or two prior to return to school, if that confidence would help the transition to the school year routine...
I was excited on Tuesday, July 16th, 2013. This was my third meeting with this student and I finally had a breakthrough with him. On the first meeting it was clear that he saw Algebra I almost as a foreign language. I began with one of the test packet, and had him do 10 questions and reviewed the questions he had done wrong. So this continued for a while, and of course sometimes he would say that he understood, but it was clear that he did not. Anyway, after reviewing the entire packet I began a teach and learn session, in which I picked a variety of topics and had him practice various equations. After which I gave him a quiz.
He failed the quiz miserably, so of course he still did not understand. Anyway, I gave him another packet for homework. When I saw the student again, I reviewed with him, but still not much improvement, but at least he tried. I did the teach and learn session again, of which some of the questions were from the previous session, and I gave him the same...
Although learning is awesome, it can be a difficult and frustrating journey for many students. This difficulty, however, is often times quite normal although most feel it means that a child may not be able to learn or that he/she is so frustrated that learning is no longer taking place. This is where the experienced tutor steps in; for frustration in learning is a part of the learning itself.
I have taught and tutored many students and have seen first hand how this frustration can leave some students, and their parents, feeling helpless and hopeless. But there is ALWAYS Hope!!! What they have failed to realize is that as the brain learns difficult concepts, it can only take in parts at a time, little parts at a time. So although it may seem no learning is taking place, it actually is, just in smaller segments. In fact, the most frustration comes right before a new concept is achieved. This is when most children become the 'most' frustrated. The may not want to go to school, complain...
Humans have a tremendous capacity to learn and adapt. However, we consistently build barriers that hinder our natural ability to change and grow. Many people, regardless of age, perceive themselves as not being talented enough to excel at math and science. They view math and science as the realms in which only scientists, engineers, mathematicians, and geniuses truly soar.
Nothing could be further than the truth. Sure, possessing a natural affinity towards these subjects helps. Yet, a supposed lack of talent does not prevent you from learning. The path may be more arduous. The journey may be longer. Nevertheless, you possess within you the fire to endure. Willpower, dedication, self belief, and an open mind can compensate for any lack of ability.
Bruce Lee was a legendary martial artist, actor, and philosopher who continues to inspire millions with the sheer intensity which he pursued his endeavors. Frail, sickly, and small as a child, Bruce Lee overcame many physical limitations...
Each summer I have a few students who work on both math and reading to keep the 'flow' and/or prep for the upcoming year. These students and their parents are completely committed to the idea of
always learning as opposed to the idea of only learning in the classroom or merely learning during the school year... in essence, the parents are setting the foundation for lifelong learning.
I would never ask a student to do work which I would not be willing to do myself or work through with them in tutoring. To this end, I have the opportunity to do reading AND catch up on my practice. This summer I am reading 'The Joy of X-A Guided Tour of Math, from One to Infinity' by Steven Strogatz at Cornell University. I LOVE this book! It is almost as good as being in a lecture or small gathering and has helped me explore how I think about math and how to share these ideas with my students.
One of my students recommended 'Hoot' by Carl Hiassen and it is on my list for the library....
Before I go run a marathon, play with my family at the pool, ride a roller coaster, head to the beach, or eat some serious amounts of ice cream, I will look back on the successful school year I have had.
I got to tutor over 15 students in Middle School, High School, SAT, and College Math...and even Chemistry! I watched GPAs rise for everybody-some were happy just to pass that College Math course to graduate, others enjoyed their hard earned As that were brought up from the D level. I must say, that things did get crazy with exams at the end of the year, but it all worked out amidst our busy-ness.
My tutoring schedule is light for the summer, and I am hoping nobody waits until December exams to contact me! I want things to be done the right way...after all of the swimming, adventure, and ice cream, of course! :)
Now that students, teachers, parents and tutors have had a chance to catch their breath from final exams, it's time to make use of the weeks we have before school starts back. Consider all that could be accomplished in the next few weeks:
Areas of math that students NEVER REALLY GRASPED could be fully explained. This could be
elementary skills like adding fractions, middle school topics like systems of equations, or
high school areas like sequences and series.
Students could have a TREMENDOUS HEAD STARTon topics that will be covered in the first few weeks of school. Imagine your son or daughter being able to raise their hand to answer a question in the first week of school because they had worked several problems just like the ones that the teacher is demonstrating.
ENORMOUS PROGRESS could be made in the area of preparation for the standardized tests (PSAT, SAT, ACT and more) that are so important to getting into a great college.
A wise man once told me: "You can continue to beat your head against that rock, but you will not chip the rock, your head (on the other hand) will be deformed." I guess I should have seen it coming, my being summarily fired from a tutoring job - The parent (in this case the mother) demanding extra "busy-work" for her son between sessions, the lack of discipline, on the student's part (especially his inability to do homework or speak to his subject teacher) and his continual lack of attention during sessions. The call came, "You are not coming here anymore, Billy Ben (not his real name) ONLY got an 81 on his Geometry test. We want top performance, 95 or better, YOU failed." Did I tell you that this student, previous to my seeing him, was working on a solid average of 40? So, it was over. Had I failed? I'm not so sure. First, I didn't take HIS test, and second, knowing the student as I did, I actually thought that an 81 was pretty good and we might have...
Hi math students :)
When preparing for a mathematics tutoring session, try to have the following things at hand...
Textbook (online or e-text)
Syllabus, assignment, tips/hints/suggestions, answer sheet/key
Pencils, pens, erasers, paper (graph paper, ruler, protractor)
All necessary formulas, laws, tables, constants, etc.
Calculator that you will use on tests
Do I really need my calculator? I can do most of my work in my head.
Having your calculator is just as important as paper and a pencil in most cases. You'll be using it on your test and if you don't know how to input what you want, you won't do very well. Have your tutor teach you about your calculator's functions beforehand. Learn how to check your simple math and how to input exponents, logarithms, or trigonometric functions before your test.
Why do I need my book, notes, or answer key? Isn't the tutor supposed to know everything?
Yes :), but even the most experienced tutor...