(iii) What valu creases? Explain. 20. Prove Theorem 4.3.1 for the case where point D lies between points 21. Redraw Fig. 4.3.2 so that A is between E and D and then prove 22. Use Heron's formula (see...

24

Extracted text: (iii) What valu creases? Explain. 20. Prove Theorem 4.3.1 for the case where point D lies between points 21. Redraw Fig. 4.3.2 so that A is between E and D and then prove 22. Use Heron's formula (see the Historical Note at the beginning of this E and F (see Fig. 4.3.2). Theorem 4.3.1. section) to determine the area of the triangle shown in Fig. 4.3.4. 26 Note: In Exercises 23-26, you may use the Pythagorean relationship (i.e. a2+b2=c2) even though this result won't be proved until the next section 23. If A ABC is a 3 cm-4 cm-5 cm right triangle with the right angle at C. show that Heron's formula gives the same area as Theorem 4.3.4 24. If A ABC is a right triangle with legs of length a and b and with hypotenuse of length c, show that Heron's formula gives the same area as Theorem 4.3.4. 25. Show an equilateral triangle in which each side has length s has area 4.4 HIST the r tribu given by A - 3 alge the


Extracted text: Figure 4.3.3 The elementary theorems listed next are derived in a similar fashion the proofs are left as exercises. Some theorems involve the use of terms he are familiar but have not yet been defined (e.g., the height of a triangle), As you consider these theorems, try to formulate valid definitions for the vocah. ulary used. Several of the exercises involve writing these definitions. Theorem 4.3.3. The area ofa right triangle is one-half the product of the lengths of its legs. Theorem 4.3.4. The area of a triangle is one-half the product of any base and the corresponding height. Theorem 4.3.5. The area of a trapezoid is the product of its height and the arithmetic mean of its bases. Theorem 4.3.6 The area of a rhombus is one-half the product of the lengths of the diagonals. In addition, several other standard area formulas concerning polygonal regions are explored as exercises in the following set.