One of the angles of an isosceles triangle measures 96°.

28. Given: Quadrilateral ABCD with

Prove: DC 7 AB AB ⬵ DE D C A E B

29. For and

not shown, suppose that and but that . Draw a conclusion regarding the lengths of and .

30. In not shown, point Q lies on

so that bisects . If , draw a conclusion about the relative lengths of and . In Exercises 31 to 34, apply a form of Theorem 3.5.10.

31. The sides of a triangle have lengths of 4, 6, and x. Write

an inequality that states the possible values of x.

32. The sides of a triangle have lengths of 7, 13, and x. As in

Exercise 31, write an inequality that describes the possible values of x.

33. If the lengths of two sides of a triangle are represented by

and in which x is positive, describe in terms of x the possible lengths of the third side whose length is represented by y. 3x + 7 2x + 5 QP NQ MN 6 MP ∠NMP MQ NP 䉭MNP EF BC m ∠A 6 m∠D AB ⬵ DE AC ⬵ DF 䉭DEF 䉭ABC

34. Prove by the indirect method: “The length of a diagonal of

a square is not equal in length to the length of any of the sides of the square.”

35. Prove by the indirect method:

Given: is not isosceles Prove:

36. Prove by the indirect method:

Given: Scalene in which bisects point W lies on . Prove : is not perpendicular to . In Exercises 37 and 38, prove each theorem.

37. The length of the median from the vertex of an isosceles

triangle is less than the length of either of the legs.

38. The length of an altitude of an acute triangle is less than

the length of either side containing the same vertex as the altitude. XY ZW XY ∠XYZ ZW 䉭XYZ PM Z PN 䉭MPN PERSPECTIVE ON HISTORY Sketch of Archimedes Whereas Euclid see Perspective on History, Chapter 2 was a great teacher and wrote so that the majority might understand the principles of geometry, Archimedes wrote only for the very well educated mathematicians and scientists of his day. Archimedes 287–212 B . C . wrote on such topics as the measure of the circle, the quadrature of the parabola, and spirals. In his works, Archimedes found a very good approximation of . His other geometric works included investigations of conic sections and spirals, and he also wrote about physics. He was a great inventor and is probably remembered more for his inventions than for his writings. Several historical events concerning the life of Archimedes have been substantiated, and one account involves his detection of a dishonest goldsmith. In that story, Archimedes was called upon to determine whether the crown that had been ordered by the king was constructed entirely of gold. By applying the principle of hydrostatics which he had discovered, Archimedes established that the goldsmith had not constructed the crown entirely of gold. The principle of hydrostatics states that an object placed in a fluid displaces an amount of fluid equal in weight to the amount of weight the object loses while submerged. ␲ One of his inventions is known as Archimedes’ screw. This device allows water to flow from one level to a higher level so that, for example, holds of ships can be emptied of water. Archimedes’ screw was used in Egypt to drain fields when the Nile River overflowed its banks. When Syracuse where Archimedes lived came under siege by the Romans, Archimedes designed a long-range catapult that was so effective that Syracuse was able to fight off the powerful Roman army for three years before being overcome. One report concerning the inventiveness of Archimedes has been treated as false, because his result has not been duplicated. It was said that he designed a wall of mirrors that could focus and reflect the sun’s heat with such intensity as to set fire to Roman ships at sea. Because recent experiments with concave mirrors have failed to produce such intense heat, this account is difficult to believe. Archimedes eventually died at the hands of a Roman soldier, even though the Roman army had been given orders not to harm him. After his death, the Romans honored his brilliance with a tremendous monument displaying the figure of a sphere inscribed in a right circular cylinder. PERSPECTIVE ON APPLICATION Pascal’s Triangle Blaise Pascal 1623–1662 was a French mathematician who contributed to several areas of mathematics, including conic sections, calculus, and the invention of a calculating machine. But Pascal’s name is most often associated with the array of numbers known as Pascal’s Triangle, which follows: 1 1 1 1 2 1 1 3

3 1

1 4

6 4

1 Each row of entries in Pascal’s Triangle begins and ends with the number 1. Intermediate entries in each row are found by the addition of the upper-left and upper-right entries of the preceding row. The row following 1 4 6 4 1 has the form 1 5 10 10

5 1

Applications of Pascal’s Triangle include the counting of subsets of a given set, which we will consider in the following paragraph. While we do not pursue this notion, Pascal’s Triangle is also useful in the algebraic expansion of a binomial to a power such as , which equals . Notice that the multipliers in the product found with exponent 2 are 1 2 1, from a row of Pascal’s Triangle. In fact, the expansion leads to , in which the multipliers also known as coefficients take the form 1 3 3 1, a row of Pascal’s Triangle. Subsets of a Given Set A subset of a given set is a set formed from choices of elements from the given set. Because a subset of a set with n elements can have from 0 to n elements, we find that Pascal’s Triangle provides a count of the number of subsets containing a given counting number of elements. a 3 + 3a 2 b + 3ab 2 + b 3 a + b 3 a 2 + 2ab + b 2 a + b 2 T T T T T T T T