Section 12-10: The Side-Splitting Theorem (Day 88)

Section 12-10 of the U of Chicago text covers the Side-Splitting Theorem. I skip to this section next because it corresponds to the next theorem in Wu.

The U of Chicago version of the theorem is:

Side-Splitting Theorem:
If a line is parallel to a side of a triangle and intersects the other two sides in distinct points, it "splits" these sides into proportional segments.

And here's Dr. Wu's version of the theorem:

Theorem 24. Let triangle OPQ be given, and let P' be a point on the ray OP not equal to O. Suppose a line parallel to PQ and passing through P' intersects OQ at Q'. Then OP'/OP = OQ'/OQ = P'Q'/PQ.

Notice that while the U of Chicago theorem only states that the two sides are split proportionally, Wu's version states that all three corresponding sides of both sides are proportional.

Moreover, the two proofs are very different. The U of Chicago proof appears to be a straightforward application of the Corresponding Angles Parallel Consequence and AA Similarity. But on this blog, we have yet to define similarity at all, much less give AA Similarity. And I jumped over those sections of this chapter because I'm following Wu -- who therefore hasn't defined similarity or AA Similarity yet. So how can Wu prove his theorem?

We've seen several examples during the first semester -- a theorem may be proved in a traditionalist text using SSS, SAS, or ASA Congruence, but these three in Common Core Geometry are theorems whose proofs go back to reflections, rotations, and translations. Instead, here we skip the middle man and prove the original high-level theorem directly from the transformations. We saw this both with the Isosceles Triangle Theorem (proved from reflections in the U of Chicago) and the Parallelogram Consequences (proved from rotations in Wu).

So we shouldn't be surprised that Wu proves his version of the Side-Splitting Theorem using transformations as well. Naturally, Wu uses dilations. In fact, the names that Wu gives the points gives the game away -- O will be the center of the dilation, and P and Q are the preimage points, while P' and Q' are the images.

Here is Wu's proof: He considers the case where point P' lies on OP -- that is, the ratio OP'/OP, which he labels r, is less than one. This is mainly because this case is the easiest to draw, but the proof works even if r is greater than unity. Let's write what follows as a two-column proof:

Given: P' on OP, Q' on OQ, PQ | | P'Q', r = OP'/OP
Prove: OP'/OP = OQ'/OQ = P'Q'/PQ

Statements                                           Reasons
1. P' on OP, Q' on OQ, PQ | | P'Q'      1. Given
2. OP' = r * OP                                    2. Multiplication Property of Equality
3. Exists Q0 such that OQ0 = r * OQ  3. Point-Line/Ruler Postulate
4. For D dilation with scale factor r,    4. Definition of dilation
    D(Q) = Q0, D(P) = P'
5. P'Q0 | | PQ, P'Q0 = r * PQ               5. Fundamental Theorem of Similarity
6. Lines P'Q0 and P'Q' are identical      6. Uniqueness of Parallels Theorem (Playfair)
7. Points Q0 and Q' are identical          7. Line Intersection Theorem
8. OQ' = r * OQ, OP' = r * OP,            8. Substitution (Q' for Q0)
    P'Q' = r * PQ
9. OP'/OP = OQ'/OQ = P'Q'/PQ = r    9. Division Property of Equality

Now the U of Chicago text also provides a converse to its Side-Splitting Theorem:

Side-Splitting Converse:
If a line intersects rays OP and OQ in distinct points X and Y so that OX/XP = OY/YQ, then XY | | PQ.

The Side-Splitting Converse isn't used that often, but it can be used to prove yet another possible construction for parallel lines:

To draw a line through P parallel to line l:
1. Let X, Y be any two points on line l.
2. Draw line XP.
3. Use compass to locate O on line XP such that OX = XP.
4. Draw line OY.
5. Use compass to locate Q on line OY such that OY = YQ.
6. Draw line PQ, the line through P parallel to line l.

This works because OX = XP and OY = YQ, so OX/XP = OY/YQ = 1.

The U of Chicago uses SAS Similarity to prove the Side-Splitting Converse, but Wu doesn't prove any sort of converse to his Theorem 24 at all. Notice that many of our previous theorems for which we used transformations to skip the middle-man, yet the proofs of their converses revert to the traditionalist proof -- once again, the Parallelogram Consequences. The Side-Splitting Converse is likewise best proved via SAS Similarity, so it will have to wait until next week.

Another difference between U of Chicago and Wu is that the former focuses on the two segments into which the side of the larger triangle is split, while Wu looks at the entire sides of the larger and smaller triangles. This is often tricky for students solving similarity problems!