Lab 6 Part II: Ratios, transversals and parallel projection
Open the GSP file lab06-v2.gsp to Page 1.
This figure looks like this, with some measurements as well. There are two lines through and the other lines are parallel to each other. We can think of this in terms of transversals, but we can also think of it in terms of parallel projection of one line to another.
Definition: Given two lines m and n in the plane, and a line PQ, the parallel projection from m to n in direction PQ is a function from m to n defined so that any point A on m is mapped to the point B on n that is the intersection of line n with the line through A parallel to line PQ.
Thus in this case, each Bk
is the image of point Ak
by parallel projection from OA to
Theorem:
Suppose points W', X', Y', Z' are points on n are parallel projections of points W, X, Y, Z on m:
a) There is a constant k so that W'X'/WX = Y'Z'/YZ = k for all W, Z, Y, Z; this k is the scaling ratio from m to n.
b) Consequently, the ratio W'X'/Y'Z' = WX/YZ for all W, Z, Y, Z (corresponding ratios on m and n are equal.
(This has mostly been proved in BG, the rest will be proved later.)
Experiment A
You can move the figure around a bit. Then stop moving the figure and record these measurements without changing any points. Notice that some ratios are equal. Think about why each is true.
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OA = |
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OB/OA = |
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OA1 = |
OB1 = |
OB1/OA1 = |
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OA3 = |
OB3 = |
OB3/0A3 |
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A1A3 = |
B1B3 = |
B1B3/A1A3 |
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A2A4 = |
B2B4 = |
B2B4/A2A4 = |
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A1A3/A2A4 = |
B1B3/B2B4 = |
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A2A4/OA = |
B2B4/OB = |
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A1B1 = |
AB = |
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OA1/OA = |
OB1/OB = |
A1B1/AB = |
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What is the scaling ratio k in this case? Which ratios = k?
What is the ratio of similarity R from triangle OAB to OA1B1? Which ratios = R?
Part B
Without moving any other points, drag point B to another position and note the new measurements.
- Which measurements stayed the same and which ones changed?
- Which one of the ratios k and R above stayed the same? Why?
- Which of the ratios in the table stayed the same? Why?
- Make two new ratio measurement using 3 points. Select IN ORDER, points O, A, A1 and then choose Measure> Ratio. Then select O, B, B1 and measure this ratio also. Notice exactly which points are in the numerator and which are in the denominator.
Part C
Now move two of the parallel lines. Move line A1B1 up so that it is on the opposite side of O from the other lines. Move line A4B4 so that it is now above line A3B3.
- Which measurements stayed the same and which ones changed?
- Notice particularly what happened to the 3-point ratios that you measure in Part B. What do you think is the reason for the difference between these ratios and the earlier OA1/OA?
- Which one of the ratios k and R above stayed the same? Why?
- Which of the ratios in the table stayed the same? Why?
Part D
Now open page 2 of the Lab 6 sketch. This has 3 lines OA,
Explain to yourself which ratios involving the Ck will be the same as these. Check some of your answers by measuring in the sketch.
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A1A/A2AO = |
B1B/B20 = |
B2B/A2A = |
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A2A/OA = |
B2B/OB = |
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OA1/OA = |
OB1/OB = |
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- Also, note the existing 3 equal ratios in the figure. Drag C so that sometimes the ratios become negative.
- Give an explanation of why these ratios are the same.
- Special Case: If C is the midpoint of A and B, what are all the ratios? How are C1 and C2 related to A1B1 and A2B2?