(Triangle ABC is circumscribed about a circle.)
P, Q, and R are tangent points. What is the perimeter of Triangle ABC?
(The following perimeter segments are given: AP = 1.2", BQ = 6.2", CQ = 4.1", while AR, BP, and CR are not given.)
Now Pappas usually gives several tangent segment problems throughout the year. Almost all of them, including this one, rely on the following theorem:
Theorem:
Two segments tangent to a circle from a common point are congruent.
This theorem appears nowhere in the U of Chicago text. The closest we get to this theorem is in Lesson 13-5 of the U of Chicago text, where we see the following Question 14:
14. Given: Point P is outside circle N;
Prove: PXNY is a kite. (You will have proved that the two tangents from circle N from point P have the same length. This is a theorem you should remember.)
In the modern Third Edition of the text, this appears in Lesson 14-4 on tangent lines. The desired theorem is given a name -- the Two Tangent Theorem -- but it's likewise buried in the exercises.
The proof isn't difficult -- it requires HL Congruence and another theorem that is highlighted in the text, the Radius-Tangent Theorem.
Proof:
Statements Reasons
1. bla, bla, bla 1. Given
2.
3. NP = NP 3. Reflexive Property of Congruence
4. NX = NY 4. Definition of circle (all radii of circle are congruent)
5. Triangle NPX = NPY 5. HL Congruence Theorem
6. PX = PY 6. CPCTC
7. PXNY is a kite 7. Definition of kite [steps 4,6]
But once again, this proof isn't highlighted in the text.
OK, so let's use the Two Tangent Theorem to solve the Pappas problem. The theorem tells us that the three missing lengths are equal to the three given lengths -- AR = 1.2", BP = 6.2", CR = 4.1". Since the total perimeter is the sum of all six lengths, we might as well double the sum of three lengths:
2(1.2 + 6.2 + 4.1) = 2(11.5) = 23
So the perimeter is 23 inches -- and of course, today's date is the 23rd.
For each Pappas problem, I want to discuss how it fits into our Geometry curriculum. Well, once again we can't give this problem today, since we won't reach tangents until Chapter 13. We'd have to emphasize the Two Tangent Theorem if we expect students to solve this problem.
Meanwhile, those who don't like how we skipped over Lesson 8-1 on perimeter can sneak perimeter problems into the rest of the curriculum. For example, today's perimeter problem can be taught during the lesson on tangents in Chapter 13. OK, that's enough -- let's get to today's actual lesson now.
Lesson 9-4 of the U of Chicago text is called "Plane Sections." In the modern Third Edition of the text, plane sections appear in Lesson 9-6. The new edition of the text makes it clear that spheres are introduced in this lesson as well.
Indeed, let's start with Euclid's definition of a sphere and related terms:
Just as with cylinders and cones, Euclid's spheres are solids of revolution. In the U of Chicago text, a sphere is the set (or locus) of all points in space a fixed distance from a point. The one term we can't define for a general sphere is its axis, unless we have a particular rotation in mind (such as the earth).
David Joyce tells us that Euclid's sphere proofs aren't as rigorous as they could be. According to Joyce, Euclid hints at the proof that a plane section of a sphere is a circle in his Book XII. A full proof appears as Exercise 20 in our text, but I chose not to include the proof on our worksheet.
Oh yes -- plane sections, the other topic of this section. It turns out that both plane sections and Euclid's solids of revolution appear in the Common Core Standards:
CCSS.MATH.CONTENT.HSG.GMD.B.4
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
We can keep this standard in mind by discussing both plane sections from today's U of Chicago lesson as well as Euclid's definition of cylinder, cone, and sphere. Naturally, I decided to slip this into the worksheet for today.
As usual, we'll look at the next proposition in Euclid:
We notice that once again, Euclid begins his proof with "For suppose it is not," which indicates that an indirect proof is needed:
Given:
Prove:
Indirect Proof:
Assume that the lines aren't coplanar -- that is,
By Point-Line-Plane, part g, the intersection of planes P and Q must be a line -- and that line can only be line EF. But now there are two lines through points E and F -- one that lies in plane P (the intersection line) and one that doesn't (the one through point G). This is a contradiction, since by Point-Line-Plane, part c, there is only one line through two points. Thus the assumption that the lines aren't coplanar is false. Therefore
According to David Joyce, Euclid assumes without proof that every line lies in a plane. Our version of the Point-Line-Postulate actually does prove that every line lies in a plane, as follows: By part b, every line contains at least two points (labeled 0 and 1). By part a, there is a (third) point in the plane not on the line. Finally by part f, through these three noncollinear points there is a plane. QED
Of course, the hidden assumptions that we all make are subtle. In fact, neither Euclid's postulates nor our Point-Line-Plane Postulate can refute the following statement:
"Space contains exactly one point (with no lines and no planes)."
This seems absurd -- space clearly contains infinitely many points. OK then, let's try to prove it by looking at each part of the postulate:
Point-Line-Plane Postulate:
a. Given a line in a plane, there exists a point in the plane not on the line. This can be written as "if a line is in a plane, then there exists a point in the plane not...." In other words, it tells what happens if a line lies in a plane, and makes no claims about what happens if there are no lines and no planes. Thus part a is (vacuously) true for single-point space.
b. Every line is a set of points that can be put into a one-to-one.... In other words, if there is a line, then it is a set of points. No claim is made if there are no lines. Thus part b is (vacuously) true for single-point space.
c. Through any two points there is exactly two line. In other words, if we have two points, then there is a line through them. No claim is made if there is only one point. Thus part c is (vacuously) true for single-point space.
And the same thing happens with parts d-g as well. There simply is no way to prove:
-- There exists two distinct points.
with no "if" or other precondition. The Point-Line-Plane Postulate has a one-element model. Yes, we did just prove that every line lies in a plane -- that is, "if there is a line, then it lies in a plane."
Notice that as soon as we have two points, then we have the intended model of Geometry. With two points, part c gives us a line passing through them. Then part b places infinitely many points on this line, one for every real number. Then from above, we know that this line lies in a plane, and so on.
This may seem like a big deal about nothing. But unless we can assert that at least two points exist, a "wise guy" student could challenge the entire Geometry course by answering every question with "point P" if it asks for a point, and "none" or "zero" if it asks for anything else (lines, planes, length, area, volume, and so on). Then the student can claim that he deserves 100% A+ in the course since neither the teacher nor the text ever refutes the statement "space contains exactly one point"!
It is Hilbert who assures us that two points exist, not Euclid or the U of Chicago. Indeed, Hilbert provides the following:
I.3 There exists at least three points that do not lie on the same line.
I.8 There exists at least four points not lying on a plane.
Hilbert specifically mentions that we don't need a line or a plane to exist in order for the trio or quartet of points to exist, since these sets aren't collinear or coplanar, respectively. Sometimes Hilbert's I.8 is written as:
"Space contains at least four noncoplanar points."
And we don't need to say, "if space exists, then ...." since space is defined as the set of all points. So space exists even if there's only one point, or even no points (the empty set). It's only for any of the undefined terms point, line, and plane where we can't automatically assume that any of them exist.
Here is today's worksheet. Notice that in today's lesson, we hint at both spherical geometry (great circle routes) as well as the conic sections of Algebra II. The Bonus Question comes from the Exploration Question and looks at the more exotic plane sections of a cube.
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