9 ## COMMON MEASUREWE SAY THAT A SMALLER MAGNITUDE measures a larger (or an equal) one if the larger magnitude is its multiple. If the straight line FG is the fourth multiple of the straight line E, we say that E measures FG four times. "Measures", in geometry, is like "is a divisor of" in arithmetic. Problem 1. In this figure, all the segments are equal. a) How many times does AB measure CE? To see the answer, pass your mouse over the colored area. Twice. b) How many times does AB measure FL? Five times. c) Does CE measure FL? No. FL is not a multiple of CE. d) Does CE measure FK? Yes. Twice. e) Does AB measure FK? Yes. Four times.
Problem 2. Borrowing the word
Problem 3. Name the first five multiples of ½ inch. ½ in, 1 in, 1½ in, 2 in, 2½ in. Problem 4. Let a straight line be 1½ meters long. a) Does a straight line ½ meter long measure it?
Yes. b) Does a straight line 1 meter long measure it? No. 1½ meters is not a multiple of 1 meter. Common measure E is a common measure of AB and CD. E measures AB two times, and it measures CD three times. If E were 1 inch, then AB would be two inches and CD would be three inches.
Thus, magnitudes have a common measure when each of them is a multiple of the same unit. What could that unit be? It could be either the unit of measure itself, such as 1 inch, or one of its unit fractions, such as ¼ inch. In particular:
For, every rational number -- every fraction -- is a multiple of some unit fraction. And 1 is a multiple of every unit fraction. (Lesson 6.) Problem 5.
Yes. 1/8 inch. b) Which multiple of the common measure is 1 inch, and which
c) Express the ratio of those lengths as a ratio of natural numbers.
Problem 6. a) What length is a common measure of 1 meter and 3.72 meters? .01 meters b) Which 1 m = 100 × .01 m. 3.72 m = 372 × .01 m. c) Proportionally, 1 meter : 3.72 meters = 100 : 372 Problem 7. If the unit of measure is 1 mile, and L is a straight line, then what are the possible common measures of 1 mile and L? 1 mile itself, or one of its unit fractions. E is a common measure of AB and CD. It measures AB three times, and CD four times. a) If E is 1 meter, then how long are AB and CD? AB = 3 m. CD = 4 m. b) If E is ½ meter how long are AB and CD? AB = 1½ m. CD = 2 m. c) If E is one-thousandth of a meter, how long are AB and CD? AB = .003 m. CD = .004 m. d) In any case, AB is to CD in the same ratio as which two natural AB : CD = 3 : 4 We can now state the following theorem:
Theorem. This should be obvious, because each magnitude will be a Again,
Problem 9. What One number is a multiple of the other, a part of it, or parts of it. Problem 10. AB, CD are straight lines, and AB is two fifths of CD. a) Illustrate that. b) Do AB, CD have a common measure? Yes. They are in the same ratio as two natural numbers. c) Which multiples of it are AB, CD? AB : CD = 2 : 5 d) If AB is 1 cm, then how long is their common measure? ½ cm How long then is CD? 5 × ½ = 2½ cm. Problem 11. EF is four times longer than GH. a) Illustrate that. b) Do they have a common measure? Yes. They are in the same ratio as two natural numbers. c) What is their common measure? GH Problem 12. Show that each of these is to 1 meter in the same ratio as two natural numbers. In each case, what length is their common measure? a) 3 meters. 3 m : 1 m = 3 : 1. Their common measure is 1 meter. b) 3¼ meters.
3¼ m : 1 m = 13/4 : 1 = 13 : 4. c) 0.512 meters.
0.512 m : 1 m = 512 : 1000. d) 9.999999 meters.
9.999999 m : 1 m = 9,999,999 : 1,000,000. Problem 13. The following three statements refer to the same fact. Do you see that?
a) Illustrate those statements when AB is 2¼ inches. 2¼ is a rational number. ¼ inch is the common measure of AB and 1 inch. AB : 1 inch = 2¼ : 1 = 9/4 : 1 = 9 : 4 b) Illustrate them when AB is .59 meters. .59 is a rational number. .01 m is the common measure of AB and 1 m. AB : 1 m = .59 : 1 = 59 : 100 Problem 14. This rectangle is three fourths of that one. Do they have a common measure of area? Yes. They are in the same ratio as two natural numbers. What magnitude is their common measure? The third part of the smaller rectangle, or, equivalently, the fourth part of the larger. Problem 15. This square is half the area of the circle. Do they have a common measure of area? Yes. They are in the same ratio as two natural numbers. If it takes one can of paint to paint the square, then it will take exactly two cans to paint the circle. Next Topic: Squares and their sides Copyright © 2017 Lawrence Spector Questions or comments? E-mail: themathpage@yandex.com Private tutoring available. |