In trapezoid RSTV, , TEST 534

PERSPECTIVE ON HISTORY Sketch of Thales One of the most significant contributors to the development of geometry was the Greek mathematician Thales of Miletus 625–547 B . C .. Thales is credited with being the “Father of Geometry” because he was the first person to organize geometric thought and utilize the deductive method as a means of verifying propositions theorems. It is not surprising that Thales made original discoveries in geometry. Just as significant as his discoveries was Thales’ persistence in verifying the claims of his predecessors. In this textbook, you will find that propositions such as these are only a portion of those that can be attributed to Thales: Chapter 1: If two straight lines intersect, the opposite vertical angles formed are equal. Chapter 3: The base angles of an isosceles triangle are equal. Chapter 5: The sides of similar triangles are proportional. Chapter 6: An angle inscribed in a semicircle is a right angle. Thales’ knowledge of geometry was matched by the wisdom that he displayed in everyday affairs. For example, he is known to have measured the height of the Great Pyramid of Egypt by comparing the lengths of the shadows cast by the pyramid and by his own staff. Thales also used his insights into geometry to measure the distances from the land to ships at sea. Perhaps the most interesting story concerning Thales was one related by Aesop famous for fables. It seems that Thales was on his way to market with his beasts of burden carrying saddlebags filled with salt. Quite by accident, one of the mules discovered that rolling in the stream where he was led to drink greatly reduced this load; of course, this was due to the dissolving of salt in the saddlebags. On subsequent trips, the same mule continued to lighten his load by rolling in the water. Thales soon realized the need to do something anything to modify the mule’s behavior. When preparing for the next trip, Thales filled the offensive mule’s saddlebags with sponges. When the mule took his usual dive, he found that his load was heavier than ever. Soon the mule realized the need to keep the saddlebags out of the water. In this way, it is said that Thales discouraged the mule from allowing the precious salt to dissolve during later trips to market. PERSPECTIVE ON APPLICATION Square Numbers as Sums In algebra, there is a principle that is generally “proved” by a quite sophisticated method known as mathematical induction. However, verification of the principle is much simpler when provided a geometric justification. In the following paragraphs, we:

1. State the principle 2. Illustrate the principle

3. Provide the geometric justification for the principle

Where , , or 9. Where , , or 16. The geometric explanation for this principle utilizes a wrap-around effect. Study the diagrams in Figure 4.41. 1 + 3 + 5 + 7 = 4 2 n = 4 1 + 3 + 5 = 3 2 n = 3 Where n is a counting number, the sum of the first n positive odd counting numbers is . n 2 The principle stated above is illustrated for various choices of n. Where , . Where , , or 4. 1 + 3 = 2 2 n = 2 1 = 1 2 n = 1

1 a

Figure 4.41 1 + 3 b 1 + 3 + 5 c Given a unit square one with sides of length 1, we build a second square by wrapping 3 unit squares around the