Notes — Chapter 5: Inequalities

To the Student. Geometric inequalities spring from this fact: The whole is greater than the part. For example, we show that the angle opposite the smaller of two sides in a triangle is congruent to a part of the angle opposite the larger side. This gives a theorem: If two sides are unequal in a triangle, the angles opposite to them are unequal in the same order. Congruence, the whole-part relationship, and the algebraic properties that begin the chapter will be important tools throughout it. When direct comparison of A and Z is difficult, a convenient intermediate quantity, say X, can often be found so that you can show A < X and X < Z. Then, you can conclude A < Z using the transitive property of inequality. Watch for this technique.

The changes in appearance as a configuration shifts or is seen from different points of view are a theme in this chapter, starting with a description of experiments on an infant’s sense relative distance and ending with Galileo’s observations of the planet Venus. The Triangle Inequality lets us work out the possibilities for the distances Galileo wondered about. These are consistent with the range of apparent sizes for Venus in its various phases. Art and architecture are concerned with the effects of changes in point of view and many examples in this chapter are drawn from these fields. In addition, you will see applications to planning kitchens and designing mechanical devices. _ til

Lesson 1. Properties of Inequality. This lesson opens discussing experiments that showed that depth perception is inborn. This leads to the algebraic properties of inequality, part of the algebra strand of this

book. The sum of positive numbers is greater than any of its summands. For Euclid this was again: the whole as greater than the part. The phrasing suggests a key idea of this chapter. Transitivity plays an equal role, as well. The other algebraic properties of inequalities. are needed for working out the consequences of various geometric conditions.

Visual effects and illusions are a theme in this chapter. Students draw figures, measure, and discover relationships that will be proven later. Applications include art and architecture.

Lesson 2. The Exterior Angle Theorem. A hoyering U. F. O. is sighted from two points on the ground giving a triangle whose geometry shifts as the U. F. O. moves. We see one relationship that seems to hold in this triangle whatever shape it has: The exterior angle as larger than either remote interior angle. Jacobs gives Euclid’s proof using congruence and the relationship of the whole to a part.

Applications in the exercises include scale drawing, measurement, and mechanical devices. In the exercises, students extend the argument given in the text to deal with the other remote angle, and they see an alternative proof of the Exterior Angle Theorem devised by Proclus.

Lesson 3. Triangle Side and Angle Inequalities. The opening painting shows how a change in point of view can dramatically alter appearances. As in Lesson 2, we see that certain geometric relationships must hold no matter how the scene is viewed.

If a triangle has unequal angles the opposite sides will be unequal in the same order. This gives the qualitative relationship between angles of a triangle and their opposite sides. The quantitative relationship is given by the Law of Sines, which appears in the lessons on trigonometry in Chapter 10, The Right Triangle.

There is opportunity for a good deal of critical thinking in the exercises. Set II begins with a discovery exercise leading to an alternative proof of the main result of this lesson. The last exercise in Set II revisits the Surfer problem from the book’s Prelude.

Lesson 4. The Triangle Inequality Theorem. The final lesson of this chapter, like the initial lesson is concerned with the perception of distance. This time we begin with Galileo’s speculations on the distance to the planet Venus. This lesson opens with five pictures of the planet Venus at various phases all shown at the same scale. The smallest disk, the full phase, is about a sixth the diameter of the new phase, the largest of these images. The geometry of the Venus, Sun, Earth system gives a triangle and the variations of that triangle motivate our discussion of the Triangle Inequality: The sum of two sides of a triangle exceeds the length of the third.

Exercises are based on examples from planetary motion, optics, and kitchen design, with the Surfer of the Prelude appearing as well.

In Euclid’s time people objected that the Triangle Inequality was obvious and did not require proof. But the interest in this theorem lies not in the fact that it is true, but in why it is true — and in the consequences we can derive from it. Using it and some basic distances, we see that Venus at its most distant point (in its full phase) is over six times the distance it was from Earth at its closest approach (in its new phase). This suggests why Venus in its full phase has an apparent size about a sixth of that of its full phase, as our pictures show.

The working out of the sizes of the orbits of the planets is a difficult technical matter. The underlying principle that we see here is that all of the measurements and observations must form a consistent picture. The variations in the apparent diameter of Venus in its various phases fit with the orbital sizes we give. Apparent diameters are measured in degrees or radians and the rough inverse proportionality to distance observed here should be expected. The reason is that, for small angles, the angle measure, the sine, and the tangent are all close to being proportional.

ll” Lines Are Determined by Distances. A corollary of the Triangle Inequality Theorem is that for three noncollinear points AB + BC > AC, so B can not satisfy the metric condition for being between A and C. Consequently, if AB + BC = AC, then B must lie on the same line determined by A and C, and B must be between A and C on that line. If we know the distances between all pairs of points, we can tell which points are collinear and which points lie between any given pair of points. Euclidean geometry can be developed using ideas of distance alone, although this approach is neither as easy nor as intuitive as ours. See, for example, Leonard M. Blumenthal’s book A Modern View of Geometry, W. H. Freeman and Company, San Francisco, 1961. _

▶ About Estimation. Inequalities are often used in estimation. For example, Archimedes trapped the circumference of a circle of diameter d = 1 between the perimeters of inscribed and circumscribed regular polygons of 6 x 24 = 96 sides. Using these perimeters, he showed that the circumference of this circle, πd = m1 = 7, lay between 370 and 3. In short, he proved that 37° yi RES 3. This was a considerable achievement for Archimedes, although a student with a scientific calculator can whip out the equivalent result in decimal form in a few minutes. That’s progress. We also have progress in estimation of 7, which is now known to an accuracy of billions of decimal places, although not by finding the perimeters of regular polygons. (See Supplemental notes for Chapter 14 for Archimedes’ calculation and see “to calculate better” in the supplemental material for Chapter 11 for high-precision 7 calculations.)

Precision calculations come with error estimates that are worked out using basic properties of inequalities.

For this reason, the study of inequalities is not a sideline, but is a central concern both for mathematics and for its applications. “