Given region R ´ S, explain The algebra method of FOIL multiplication is illustrated Given region

Perimeter of a Polygon Semiperimeter of a Triangle Heron’s Formula Brahmagupta’s Formula Area of Trapezoid, Rhombus, and Kite Areas of Similar Polygons Perimeter and Area of Polygons 8.2 KEY CONCEPTS TABLE 8.1 Perimeter of a Triangle Scalene Triangle Isosceles Triangle Equilateral Triangle Table 8.1 summarizes perimeter formulas for types of triangles, and Table 8.2 sum- marizes formulas for the perimeters of selected types of quadrilaterals. However, it is more important to understand the concept of perimeter than to memorize formulas. See whether you can explain each formula. We begin this section with a reminder of the meaning of perimeter. The perimeter of a polygon is the sum of the lengths of all sides of the polygon. DEFINITION a b c P = a + b + c b s s P = b + 2s s s s P = 3s TABLE 8.2 Perimeter of a Quadrilateral Quadrilateral Rectangle Square or Rhombus Parallelogram b d a c P = a ⫹ b ⫹ c ⫹ d h b b h P = 2b + 2h or P = 2b + h s s s s P = 4s b s s b P = 2b + 2s or P = 2b + s A C D B Figure 8.17 12 ft ? ? 12 ft 20 ft 18 ft Figure 8.18 EXAMPLE 1 Find the perimeter of shown in Figure 8.17 if: a AB ⫽ 5 in., AC ⫽ 6 in., and BC ⫽ 7 in. b Altitude AD ⫽ 8 cm, BC ⫽ 6 cm, and Solution a = 18 in. = 5 + 6 + 7 P ABC = AB + AC + BC AB ⬵ AC 䉭ABC 쮿 Exs. 1–4 HERON’S FORMULA If the lengths of the sides of a triangle are known, the formula generally used to calcu- late the area is Heron’s Formula named in honor of Heron of Alexandria, circa A . D . 75. One of the numbers found in this formula is the semiperimeter of a triangle, which is defined as one-half the perimeter. For the triangle that has sides of lengths a, b, and c, the semiperimeter is . We apply Heron’s Formula in Example 3. The proof of Heron’s Formula can be found at our website. s = 1 2 a + b + c We apply the perimeter concept in a more general manner in Example 2. b With , is isosceles. Then is the bisector of . If BC 6, it follows that DC 3. Using the Pythagorean Theorem, we have Now . NOTE: Because x ⫹ x ⫽ 2x, we have . 173 + 173 = 2173 P ABC = 6 + 173 + 173 = 6 + 2173 L 23.09 cm AC = 173 64 + 9 = AC 2 8 2 + 3 2 = AC 2 AD 2 + DC 2 = AC 2 = = BC ⬜ AD 䉭ABC AB ⬵ AC EXAMPLE 2 While remodeling, the Gibsons have decided to replace the old woodwork with Colonial-style oak woodwork. a Using the floor plan provided in Figure 8.18, find the amount of baseboard in lin- ear feet needed for the room. Do not make any allowances for doors b Find the cost of the baseboard if the price is 1.32 per linear foot. Solution a Dimensions not shown measure 20 12 or 8 ft and 18 12 or 6 ft. The perimeter, or “distance around,” the room is b The cost is . 76 1.32 = 100.32 12 + 6 + 8 + 12 + 20 + 18 = 76 linear ft 쮿