"I have resolved to quit only abstract geometry, that is to say, the consideration of questions which serve only to exercise the mind, and this, in order to study another kind of geometry, which has for its object the exploration of the phenomena of nature." -- Descartes
And of course the author of this quote needs no introduction. When it's time to graph equations, the first thing we think about is the coordinate or Cartesian plane, which is named after Rene Descartes, the 17th century French mathematician.
There are many very interesting topics in this chapter. First Kline looks at the equations of circles, derived from the Pythagorean Theorem, just as in Lesson 11-3 of the U of Chicago text. Then he writes about the derivation of the equation of a line:
"Had we considered a straight line inclined more steeply to the horizontal, for example, one that rises 2 units for each horizontal difference of 1 unit, then from similar triangles OQ'P' and OQP we might have argued that y/x = 2/1, or that y = 2x is the equation of the line."
So Kline is deriving the equation of a line given its slope using similar triangles -- which is also expected in the eighth grade Common Core Standards.
Then Kline writes about a line like y = 2x + 3, which is obtained from the graph of y = 2x simply by moving each point up three units -- that is, by performing a translation. And of course a line and its translation image are parallel, so the graphs of y = 2x and y = 2x + 3 are parallel.
Then Kline moves on to parabolas, including their derivation from the definition of parabola as the locus of all points equidistant from a point (the focus) and a line (the directrix).
Question 10 of the PARCC Practice Exam is on dilations:
10. The figure shows line segment
Suppose that line segment
A. Line segment
B. Line segment
C. Line segment
D. Line segment
This question is simple if you know the theorem -- a line is parallel to its dilation image. This means that the correct answer is A.
That a line is parallel to its dilation image is one of the two big properties of dilations proved in the first three lessons of Chapter 12 -- the other being Dilation Distance.
Recall that I don't like the U of Chicago proof of Dilation Distance because, according to the Common Core Standards, it's circular. We should be using dilations to prove the properties of the coordinate plane, not using coordinates to prove the properties of dilations!
I also point out that it's good to look at all of the transformations we have learned, and see which ones have lines that are parallel to their images, as well as lines that are their own images (invariant lines):
Reflections:
Invariant lines -- the mirror itself, any line perpendicular to mirror
Lines parallel to images -- any line parallel to mirror
180-Degree Rotations:
Invariant lines -- any line through the center
Lines parallel to images -- any other line
Translations:
Invariant lines -- any line parallel to direction of slide
Lines parallel to images -- any other line (as Kline does in Chapter 10)
Glide Reflections:
Invariant lines -- the mirror itself
Lines parallel to images -- any line parallel or perpendicular to mirror
Dilations:
Invariant lines -- any line through the center
Lines parallel to images -- any other line
Recall that a fixed point is a point that is actually mapped to itself. Every line on the mirror of a reflection is a fixed point, as is the center of a rotation or dilation. Translations and glide reflections have no fixed points.
Notice that 180-degree rotations (also called inversions), translations, and dilations all have a property in common -- every line is either invariant or parallel to its image. As it turns out, we can prove a theorem about all three transformations simultaneously. (In some ways, a 180-degree rotation or inversion is also a dilation with scale factor -1.)
Theorem:
Let T be a transformation preserving betweenness and collinearity, and such that through any point, there is an invariant line, and such that no fixed point lies on a line that is not invariant. Then any line not invariant is parallel to its image.
Indirect Proof:
Let l be any line that is not invariant -- that is, the image of l is not l. We are to show that l must be parallel to its image l'. So assume the contrary, that l and l' are not parallel -- that is, that they intersect at some point P. Every point on l' has an preimage on l. In particular, the preimage of P is on l -- call that point Q (that is, Q' is P). Since by hypothesis no fixed point can lie on l, so Q is not P.
By hypothesis, every point lies on some invariant line. In particular, Q lies on some invariant line -- call that invariant line q. Since l is not invariant, l is not q. By the definition of invariant, the image of every point on q is another point on q. In particular, the image of Q is on q. But the image of Q is P, so P is on q.
So notice that Q and P are both on q -- but notice that Q and P are both on l. In other words, we have two distinct lines, q and l, intersecting at two distinct points Q and P. This is a contradiction, since two lines can intersect at most one point. Therefore l | | l'. QED
PARCC Practice EOY Question 10
U of Chicago Correspondence: Lesson 12-3, Properties of Size Changes
Key Theorem:
A line and its image under a size transformation are parallel.
Common Core Standard:
CCSS.MATH.CONTENT.HSG.SRT.A.1.A
A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
Commentary: The U of Chicago text doesn't give a satisfactory proof of the this theorem, nor many problems based on the theorem, but it does mention the theorem. The Common Core Standards only emphasize that a line and its dilation image are parallel, but there's no reason why we can't look at other lines and their images -- recall the Line Parallel to Mirror and Line Perpendicular to Mirror Theorems here on the blog.
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