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[The original GSP Java "applets" expired sometime since last spring. These will be redone in time for the spring 2013 term.]
Size matters. Is one item larger, smaller, or the same size as another? If larger or smaller, how much larger and how much smaller? Is the program larger or smaller than the memory available? Is your shoe larger or smaller than your foot? Are the books larger, smaller or the same size as the backpack? How much? The answer to such questions is explained by an understanding of ratio, measure, and proportion. Understanding these three concepts turns out to be far more important than just a question of high school geometry, so it is worth the effort to clearly understand them now.
[Most of the diagrams for this page are in the frame at the right. To return to the initial page of diagrams click here.]
Socrates in Plato's Republic argues that education begins with sorting out "larger, smaller, and equal." Click on the image of Plato to read Socrates' words in the frame at the right.
Here are two triangles: one larger and one smaller. You can drag the vertices of DABC to change its shape and the length of its sides. The sides of DPQR will always remain double the size of sides DABC. The two triangles are said to be similar and their sides are said to be proportional. This site will help you understand these concepts.
The three goals of this site are to explain:
1. Ratio as a comparison of two magnitudes.
2. Measure as a way of counting how much of a magnitude.
3. Proportion as a comparison of ratios and the relation between parts of similar triangles.
[This site owes its original idea to "an unusual and attractive edition of Euclid was published in 1847 in England, edited by an otherwise unknown mathematician named Oliver Byrne. It covers the first 6 books of Euclid's Elements of Geometry, which range through most of elementary plane geometry and the theory of proportions. What distinguishes Byrne's edition is that he attempts to present Euclid's proofs in terms of pictures, using as little text - and in particular as few labels - as possible. What makes the book especially striking is his use of colour. Incidentally, at the time of its publication the first 6 books, which are the ones concerrned with plane geometry, made up the basic mathematics curriculum for many students." Byrne's Euclid]
In the 1826 an English mathematician and logician named DeMorgan wrote a book about the "connection of magnitude and number", and in that book he coined the term "quantuplicity." Quantuplicity is the answer to the question "how much?" The term never caught on, but I have always liked it. Quantuplicity is the repetition of a unit. When we ask the question "how much?" the answer makes use of quantuplicity. What is the quantuplicity of the desks in the room = how many times can we find a "desk-thing" in the room = "how many desks are there?" And how many desks are there in the next room? We can compare these two amounts and the comparison is expressed as a ratio.
The comparison of two items in terms of how much (quantuplicity) is called a ratio. Euclid's definition is the best short explanation.
The first box shows two columns, A and B. We can compare their size and see that A is longer than B. The second box shows us how much longer A is by laying B alongside A. In the second box, we use B to measure A. The third box shows the ratio A/B is just the comparison of the two lengths.
In the next example the first box has two squares, R and Q. We can see that Q can measure square R by the lines marked in R.
In the above two examples we used the smaller part to measure the larger part and that gave us an answer to the important question "how much?" The use of a measure gives us a number answer ("quantuplicity), so we can now write the ratios with numbers.
[Hint: It will help if you always think about ratios as comparisons of two things as the colored figures in the above diagrams. For now, never think of a ratio as a fraction. These are not fractions; they are ratios. Proportions are not fractions; they are ratios.]
Now look at the following figures.
Compare W to X and Y to Z as we compared R and Q above. Is the ratio of W to X the same as the ratio of Y to Z? Yes, for in each case, the smaller figure measures the larger figure two times. This gives us the following the situation pictured in the next box.
As Euclid says when he defines proportion:
So we can say that W and
X are proportional
to Y and Z.
A proportion is a comparison of two ratios that are the
same.
A proportion is a relation of four magnitudes.
| Each of these properties is illustrated in the frame at the left. |
There are more properties of proportions, but you can work them out with your own diagrams.
Our next question is: how does this relate to Geometry? The answer, of course, is that we can have figures which are the same but come in different sizes (like the squares used earlier). Figures of the same kind but of different sizes are called similar. Similar figures have their sides in proportion. For example, the two triangles in the next figure are similar and their sides are in proportion. The red triangle is a 3-4-5 triangle and the blue triangle is a 6-8-10 triangle.
We can look at two similar triangles. Each side of the larger triangle is twice the corresponding side of the smaller triangle. They are more flexible than the first pair. You can change either one at will and see that since the sides stay in the same proportion, the triangles remain similar.
The next diagram has DABC with segment DE connecting the midpoints of sides AB and CB . Change the sides and angles of the triangle and see how the measures given of certain angles, sides, and ratios remain the same. Given the measures and the ratios, which theorems or postulate can you use to prove the triangles similar?
The diagram shows DABC with an angle bisector for B. Note that the angle bisector (BD) cuts the opposite side (BC) into segments (BD, CD) in the same ratio as the two segments forming B (BA, CA). Move point A to try different sized and shaped triangles.
How would you prove this? That is, given that B is bisected by BD, prove that DABD ~ DACD.
Suppose we cut a line segment, AB, into two parts at point C. We then have the two magnitudes AC and CB in a ratio. We can construct another line segment EC so that the three segments (AC, CE and CB) are in a proportion such that
To see how the mean proportional is found by geometrical means, consult the diagram. Move point C along AB to see how the section EC remains the mean proportional. At what point are all three segments congruent?
Then there is a golden rectangle whose sides are in the "extreme and mean" ratio achieved by cutting a line segment such that the shorter segment is to the longer segment as the longer segment is to the original uncut segment.In the diagram, DF has been cut at C into the golden ration, that is,
Algebraically,
This ratio has many interesting uses and appearances. It also has many websites (e.g. the heading). The most famous and frequent use is to form the Golden Rectangle.
The diagram shows how a Golden Rectangle is constructed from a square. Given ABCD, take the midpoint (E) of one side and draw a line segment to an opposite vertex (B). Construct a circle with this radius (EB) and extend a side (DC) to the circle (DF). Complete the rectangle ADFG.
Note now that the side of the original square (AB) is the mean proportional between the long side of the rectangle (AG) and the short addition (BG).
Note also that BGFC is a Golden Rectangle.