Wednesday, November 1, 2017

Lesson 5-4: Properties of Kites (Day 54)

This is what Theoni Pappas writes on page 305 of her Magic of Mathematics:

"Two of Sam Loyd's Checkerboard Puzzles...."

This is the last page of the checkerboard subsection. These puzzles were devised by Sam Loyd, a famous 19th century chess player and puzzlist. These puzzles are visual, so let's provide a link to both of these:

This is known as Mrs. Pythagoras' Puzzle. We have a 5 * 5 square next to a 12 * 12 square and we are to cut it into three pieces in order to make a 13 * 13 square.

This one is Battle Royal. The eight pieces are to be put together in order to form an 8 * 8 square -- the usual chessboard shape.

As usual, the answers will be tomorrow. Technically, the link to Battle Royal above already has a link to the answers. Mrs. Pythagoras also has a posted solution, but it's so much different from what Pappas writes at the back of her book.

Meanwhile, here's the answer to yesterday's "The Squares of a Checkerboard Puzzle". There are:

1 square of size 8 * 8
4 squares of size 7 * 7
9 squares of size 6 * 6
16 squares of size 5 * 5
25 squares of size 4 * 4
36 squares of size 3 * 3
49 squares of size 2 * 2
64 squares of size 1 * 1
for a total of 204 squares.

Lesson 5-4 of the U of Chicago text is called "Properties of Kites." In the modern Third Edition of the text, kites appear in Lesson 6-5.

Two years ago, I changed the way I taught this lesson, and just like Lesson 5-1, I wish to revert to the original lesson. And so this is what I wrote three years ago about today's topic:

Section 5-4 of the U of Chicago text covers kites. The kite is a relatively new quadrilateral classification. Not only did Euclid never define kite, but many texts made no mention of kites -- including my class geometry textbook from 20 years ago. Nowadays most texts define kite, but some include kites only in bonus questions, not in the main text.

Here's what John Conway wrote about the kite -- over 20 years ago, right around the time that I was taking my geometry class:

In fact it's not quite true, either, because "kite" is not
a very traditional name - it was obviously inserted because
this was a type of quadrilateral that SHOULD have received a
traditional name, but didn't, until recently.

Why do we include the kite - plainly because it represents
the one type of symmetry not otherwise mentioned. But this
reason suggests we should also EXCLUDE the non-isosceles

David Joyce, meanwhile, doesn't find kites to be necessary at all:

Too much is included in this chapter. The sections on rhombuses, trapezoids, and kites are not important and should be omitted.

Notice that both Conway and Joyce want to exclude trapezoids -- but Joyce is the only writer I know who wants to omit rhombuses. Of course, for Joyce, the emphasis should be on triangles and parallel lines, not quadrilaterals like rhombuses. Also, notice that kites are not specifically mentioned in the Common Core Geometry Standards. But I find that students can identify a kite more readily than a rhombus or trapezoid.

Now every kite contains two special vertices, known as its "ends." As defined by the U of Chicago:

"The common vertices of the equal sides of a kite are the ends of the kite."

Notice that Michael Serra doesn't define ends, but does give a name to the two angles whose vertices are the ends of the kite -- the vertex angles, in analogy with the vertex angle of an isosceles triangle. I see that from a proof standpoint, this makes sense, since the first thing that we do in the proof of our main theorem is divide the kite into two isosceles triangles.

And now here is our main theorem: the Kite Symmetry Theorem. As I mentioned back in the lesson on isosceles triangles, we use symmetry to determine the properties of kites. (A pre-Common Core proof might divide the kite into two triangles to be proved congruent by SSS.) I will post a proof here of the Kite Symmetry Theorem. It is taken directly from the U of Chicago, except that I, as always, add a Given step to the beginning of the proof. Since the U of Chicago's proof has eight steps, mine has nine:

Kite Symmetry Theorem:
The line containing the ends of a kite is a symmetry line for the kite.

Given: ABCD is a kite with ends B and D.
Prove: Line BD is a symmetry line for ABCD.

Statements                                           Reasons
1. ABCD is a kite with ends B and  1. Given
2. AB = BCAD = DC                         2. Definition of ends of kite (meaning)
3. Tri. ABC and ADC are isosceles      3. Definition of isosceles triangle (sufficient)
4. Let m be the perp. bis. of AC          4. A segment has exactly one perp. bisector
5. A' = CC' = A                                  5. Definition of reflection (sufficient)
6. m contains B and D                         6. The perp. bis. of the base = angle bis. of the vertex angle
                                                                 (so it contains the vertex)
7. B' = BD' = D                                 7. Definition of reflection (sufficient)
8. ABCD reflected over m is CBAD     8. Figure Reflection Theorem
9. m (Line BD) is a symmetry line      9. Definition of symmetry line (sufficient)

Notice that more than half of the reasons in this proof are definitions. This underlines how important definitions are to the study of quadrilaterals.

As for the other theorems in this lesson, the Kite Diagonal Theorem follows directly from lines 4 and 6 of the above proof. It makes the symmetry diagonal the perpendicular bisector of the other diagonal and so the diagonals of a kite are perpendicular, and the symmetry diagonal bisects the other one.

Finally, we have the Rhombus Symmetry Theorem. It states that a rhombus has two symmetry lines, as both of the lines containing its diagonals are symmetry lines. This follows directly from the classification of a rhombus as a kite. In texts that define kite exclusively, the theorems "The diagonals of a kite are perpendicular" and "the diagonals of a rhombus are perpendicular" are two separate theorems, often in two separate sections. But here we can easily see why the diagonals of both the kite and the rhombus are perpendicular -- because the rhombus is a kite! Also, since we are defining kite inclusively, we don't need any extra steps in the proof to ensure that our kite isn't a rhombus.

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