Thursday, April 7, 2016

Lesson 13-5: Tangents to Circles and Spheres (Day 135)

Our unit on circles and spheres continues with Lesson 13-5, Tangents to Circles and Spheres. The key theorem of this lesson is:

Radius-Tangent Theorem:
A line is tangent to a circle if and only if it is perpendicular to a radius at the radius's endpoint on the circle.

Even though I created a worksheet for it, I actually didn't write much about this lesson last year. This is mostly because of the mixed-up way that I covered Chapter 13 both years on the blog.

Last year, I officially combined Lessons 13-5 and 13-6, "Uniqueness." But the worksheet that I made is almost all from 13-5, with very little from 13-6. Indeed, I already covered 13-6 this year during the first semester, as part of the unit on glide reflections -- because a theorem on glide reflections is proved in 13-6. That day, I created a brand new worksheet just for 13-6.

As I mentioned all this year, the material in Chapter 13 naturally fits in various units. The first four sections of Chapter 13 are all on logic, which much of the material being covered in Chapter 1 or 2 in many other texts. Lessons 13-7 and 13-8 are on exterior angles, and so it also fits in the first semester when we were covering triangles and other polygons. This is why we've already covered most of Chapter 13 except for today's 13-5, since it fits with our circle and sphere unit.

Recall that next year, I plan on covering the U of Chicago text in order. This way I will be respecting the order the authors have written the text rather than making up my own order. The plan is for me to follow the digit pattern with the chapter numbers and the school year day count. So I'll teach Lessons 13-1 to 13-8 on Days 131 through 138, and then give a Chapter 13 Review and Chapter 13 Test on Days 139 and 140. (This year there is no separate Chapter 13 Test.) Notice that, as luck would have it, I'm already following the digit pattern for today's lesson only, with Lesson 13-5 on Day 135.

The reason that the Radius-Tangent Theorem is given in Chapter 13 of the text is that an indirect proof is required. Here's a statement of the forward direction of the theorem and its proof as given in the text. As is typical for indirect proofs, it is given in paragraph form:

If a line is perpendicular to a radius of a circle at the radius's endpoint on the circle, then it is tangent to the circle.

Given: Circle O, OP is perpendicular to l.
Prove: l is tangent to the circle.

Indirect Proof:
Assume Q is another point on l and on the circle. Since Q is on l, Triangle OPQ is a right triangle with hypotenuse OQ. So OQ > OP. But since Q is on the circle, OQ = OP. The statement OQ > OP, and OQ = OP is a contradiction. By the Law of Indirect Reasoning, the assumption must be false. So l intersects the circle at exactly one point. By the definition of tangent (sufficient condition), l is tangent to Circle O. QED

To prove the converse, the U of Chicago text officially proves the contrapositive of the converse (that is, the inverse) instead. Recall that the converse and inverse of any conditional are equivalent. The U of Chicago text gives a paragraph proof, but this one naturally lends itself to a two-column proof:

If a line is tangent to a circle, then it perpendicular to the radius drawn to the point of tangency.

Given: OP is not perpendicular to m.
Prove: m is not tangent to circle O at point P.

Statements                               Reasons
1. OP not perp. to m                 1. Given
2. Q on m s.t. OQ perp. to m     2. Uniqueness of Perpendiculars Theorem
3. R on m s.t. R-Q-P, QR = QP 3. Point-Line-Plane Postulate (Ruler Postulate)
4. Angle OQR = OQP               4. Def. of perp. (meaning), all rt. angles congruent
5. OQ = OQ                             5. Reflexive Property of Congruence
6. Triangle OQR = OQP            6. SAS Congruence Theorem (steps 3, 4, 5)
7. OR = OP                              7. CPCTC
8. R on circle O                        8. Definition of circle (sufficient condition)
9. m, circle O intersect @ 2 pts. 9. Definition of intersect (sufficient condition)
10. m is not a tangent               10. Definition of tangent (insufficient condition)

Notice that here I used the abbreviation R-Q-P for "Q is between R and P." I abbreviated so that the two columns would fit on the blog. It's interesting that this abbreviation often appears in college Geometry texts, but never in high school Geometry texts! We could have also written step 3 as, "Locate R on m so that Q is the midpoint of RP."

By the way, here's one theorem that doesn't appear in the U of Chicago text, even though it follows directly from the Radius-Tangent Theorem that we just proved. It does appear in other texts:

If two tangents to a circle intersect at a point, then the point of intersection is equidistant from the two points of tangency.

Given: Lines AP, AQ tangent to circle O
Prove: AP = AQ

Statements                                Reasons
1. AP, AQ tangent to circle O     1. Given
2. AP perp. OP, AQ perp. OQ    2. Radius-Tangent Theorem
3. Angle APO, AQO rt. angles    3. Definition of perpendicular (meaning)
4. AO = AO                               4. Reflexive Property of Congruence
5. OP = OQ                              5. Definition of circle (meaning)
6. Triangle APO = AQO             6. HL Congruence Theorem
7. AP = AQ                               7. CPCTC

Before I give the worksheet -- the only part of last year's post that I'm putting here again today -- let me comment on the "review questions" from this worksheet. These are review questions for Chapter 13 -- most of which we covered during the first semester. So technically these questions are review for the students, but it's been several months since they've seen any of these.

One of these questions, though, deserves special attention:

8. Give an indirect proof to show that sqrt(39,600) is not 199.

In some ways, this actually is a review from Monday's lesson on Square Root Day -- depending on what square root lessons we gave that day. The indirect proof should look like this:

Indirect Proof:
Assume that sqrt(39,600) = 199. Squaring both sides gives 39,600 = 39,601, a contradiction. Thus sqrt(39,600) doesn't equal 199. QED

Notice that for all the hoopla surrounding Square Root Day, there was actually very little discussion of irrational square roots, even though the square roots of most whole numbers are irrational. This is, of course, because Square Root Day falls on dates like 4/4/16, reflecting sqrt(16) = 4, and other square roots of perfect squares -- not irrational square roots.

But it may be a good idea to continue Square Root Week with a discussion of irrational square roots, including why these square roots are irrational. Notice that the U of Chicago text continues to give questions like these throughout Chapter 13, starting in Lesson 13-4 on Indirect Proof and reviewing them all the way up to the SPUR Chapter Review. The questions from the text in that review are:

20. Give an indirect proof to show sqrt(2400) is not 49.
21. Give an indirect proof to show that sqrt(2) is not 239/169.

And notice that Question 21 here can generalized to:

-- Give an indirect proof to show that sqrt(2) is not a/b for any whole numbers a and b.

...that is, show that sqrt(2) is irrational! I remember when I first saw the famous indirect proof of the irrationality of the sqrt(2) -- it was given in a sidebar in my old Algebra I text. I had heard for a while that sqrt(2) is irrational, but I didn't realize that it was something that could be proved. The proof goes something like this:

Indirect Proof:
Assume that sqrt(2) is rational -- that is, sqrt(2) = a/b for some whole numbers a and b. Just as in all the other square root indirect proofs, we square both sides to get 2 = a^2 / b^2. Multiplying gives us the equation a^2 = 2b^2. So a must be even as its square is even -- that is, a = 2c for some other whole number c. Substituting gives us (2c)^2 = 2b^2, or 4c^2 = 2b^2, or 2c^2 = b^2. So b must be even as its square is even. As a and b must both be even, a/b can't be written in lowest terms -- but this is a contradiction as every fraction has a lowest-terms form. Therefore sqrt(2) is irrational. QED

Back on Square Root Day, I mentioned how we can come up with special activities. As it's still Square Root Week, here's an interesting one that involves irrational square roots:

Suppose you were in charge of creating an annual Square Root Day -- one that would be celebrated every year, not just once a decade or so. Since Pi Day is based on the digits of pi = 3.14..., you come up with the idea of using the digits of an irrational square root.

Question 1. What possible dates are there for Square Root Day?

Answer 1. We begin to calculate some square roots:

sqrt(1) = 1 (rational)
sqrt(2) = 1.41... (January 41st doesn't exist!)
sqrt(3) = 1.73... (January 73rd doesn't exist!)
sqrt(4) = 2 (rational)
sqrt(5) = 2.23... (February 23rd)

So we have February 23rd as our first possible Square Root Day. To find the next Square Root Day, we must go all the way up to sqrt(10) = 3.16..., or March 16th. But as it turns out, a second Square Root Day in March is possible, as sqrt(11) = 3.31..., or March 31st.

Here is a complete listing of all the possible Square Root Days:

February: 23rd (sqrt(5))
March: 16th (sqrt(10), 31st (sqrt(11))
April: 12th (sqrt(17)), 24th (sqrt(18))
May: 9th (sqrt(26)), 19th (sqrt(27)), 29th (sqrt(28))
June: 8th (sqrt(37)), 16th (sqrt(38)), 24th (sqrt(39))
July: 7th (sqrt(50)), 14th, 21st, 28th (sqrt(53))
August: 6th (sqrt(65)), 12th, 18th, 24th, 30th (sqrt(69))
September: 5th (sqrt(82)), 11th, 16th, 21st, 27th (sqrt(86))
October: 4th (sqrt(101)), 9th, 14th, 19th, 24th, 29th (sqrt(106))
November: 4th (sqrt(122)), 9th, 13th, 18th, 22nd, 26th (sqrt(127))
December: 4th (sqrt(145)), 8th, 12th, 16th, 20th, 24th, 28th (sqrt(151))

So we see that there are 44 possible Square Root Days every year. But notice that these 44 dates are not equally distributed across the months -- January has none, while December has seven. So this leads to out next question:

Question 2. Why is it that there are more Square Root Days in the later months?

Answer 2. This is a great Calculus question! I mentioned back on Monday that the function y = x^2, and its derivative y' = 2x, govern the distribution of square roots. The derivative is the slope of the tangent line (here we go with tangent lines again). This tells us that if (x, y) is a point on the graph of y = x^2, then (x + 1/m, y + 1) is on the tangent line that approximates the graph near (x, y). That is, we have that (x + 1/m)^2 is approximately y + 1, so sqrt(y + 1) is approximately x + 1/m = x + 1/(2x).

So, for example, sqrt(5) is approximately 2 + 1/4, and sqrt(10) is approximately 3 + 1/6, and sqrt(11) is approximately 3 + 1/3. Since there are at most 31 days in any month, we can approximate the number of square root days in a month by calculating 0.31(2x) = 0.62x. So February has around 0.62(2) = 1.24 square root days, while December has around 0.62(12) = 7.44 square root days -- and this is exactly what we find.

Notice that March is expected to have 0.62(3) = 1.86 square root days. It actually has two -- but we can see that March 31st just barely makes it within the month of March. This is because our earlier estimate of sqrt(11) = 3 + 1/3 is always an overestimate. This is because m = 2x from the derivative, but notice that the derivative is always increasing as x increases. So m is always an underestimate, thereby making 1/m an overestimate. So the sqrt(11) is not 3 + 1/3 = 3.33... (March 33rd doesn't exist), but is actually 3.31... (March 31st exists).

So we need to round our estimate 0.62(3) = 1.86 up to two. Most of the time though, we'll need to round down as the next Square Root Day is out of range, so for January, 0.62(1) = 0.62 must round down to zero, and for June, 0.62(6) = 3.72 must round down to three.

We also notice that the Square Root Days for some months follow an obvious pattern.

-- In April, the Square Root Days are multiples of twelve.
-- In June, the Square Root Days are multiples of eight.
-- In July, the Square Root Days are multiples of seven.
-- In August, the Square Root Days are multiples of six.
-- In December, the Square Root Days are multiples of four.

Question 3. Why do the Square Root Days in these five months follow a pattern?

Answer 3. The slope m = 2x tells us that sqrt(y + n) is approximately is x + n/(2x). Since the date of Square Root Day is based on the hundredths place, this gives us (100n)/(2x) = 50(n/x) as the date of the nth Square Root Day in the xth month.

Let's try this formula for May -- x = 5, so 50(n/x) = 10n. So the Square Root Days in May ought to be on the 10th, 20th, and 30th -- but they aren't. To see why, recall that x + n/(2x) is always an overestimate of sqrt(y + n). So sqrt(26) is not 5 + 1/10 or 5.100... but 5.099..., which equates to May 9th using strict truncation. We can compensate for this overestimate by using 48(n/x) or 49(n/x) instead of 50(n/x). For the month whose numbers that are factors of 48 (4, 6, 8, 12) or 49 (7), these formulas give the exact Square Root Days. On the other hand, Square Root Days that are later in the month are even farther away from the 50(n/x) overestimate, which is why February's lone Square Root Day is on the 23rd, rather than the 25th or even the 24th.

This leads to our final question:

Question 4. Which of the 44 possible Square Root Days should we actually celebrate in class?

Answer 4. We should choose a Square Root Day that is during the school year, so that rules out all of the July dates (unless it's a summer school class). November 22nd and 26th are likely to fall during Thanksgiving break (especially at schools that take a full week off for the holiday), and likewise December 24th and 28th fall during winter break at virtually every school.

With so many dates to choose from, we should be able to avoid the weekend. It's possible that the four July dates -- being seven days apart -- may all fall on the weekend, but we've already ruled out July anyway. It may be appealing to choose one of the early December dates -- the 4th and 8th can't both fall on the weekend, so we can choose either one depending on the year.

On the other hand, perhaps we choose a Square Root day as early in the year as possible -- sqrt(5) is much more likely to appear in real-life calculations than sqrt(151), so try February 23rd. This would mean there's a party just as the Long March is beginning. But this date is tricky -- some schools take longer breaks for Presidents Day. And suppose February 23rd falls on a Saturday. Should we wait until March 16th -- which would also fall on the weekend (Saturday without a Leap Day, Sunday with it)? Or should we try March 31st (a Sunday)? Next would be April 12th (a Friday, but possibly during spring break).

And we should rule out March 16th anyway, as this is two days after Pi Day. Under the Common Core Standards, students learn about irrational numbers -- both pi and various surds -- around the seventh or eighth grade, so in March they might as well celebrate Pi Day, not sqrt(10) Day.

If I were teaching a class and wanted to give a Square Root Day party every year, I'd probably throw it whenever we reached the unit on square roots -- hopefully this wouldn't be January. Another alternative would be to give a party during Common Core testing -- very little learning occurs during the short periods after the test anyway. There are five possible dates in April and May to choose from.

By the way, in classes below Calculus, we can give this same problem using finite differences instead of derivatives. The differences between consecutive squares are the odd numbers, which we prove using (x + 1)^2 = x^2 + 2x + 1. Using linear interpolation gives us 100n/(2x + 1) as the nth Square Root Day of the xth month. This usually ends up being a slight underestimate -- for example, it actually gives 3 + 1/7 (which is 22/7, or approximately pi) as sqrt(10). But for December we have that 100n/(2x + 1) = 4n, so it gives the exact Square Root Days for December.

Well, I'm not actually having a Square Root party today -- but still this was an interesting question from a mathematical standpoint!

Here is the worksheet on tangents to circles, with a review question on square roots:

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