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Section 8.2 Perimeter and Area of Polygons

Section 8.2 Perimeter and Area of Polygons. The perimeter of a polygon is the sum of the lengths of all sides of the polygon. Table 8.1 and 8.2 p. 363 Ex 1,2 p. 364. Theorem 8.2.1 Heron’s Formula for the Area of a Triangle. Semiperimeter s = ½ (a + b + c) Heron’s Formula:

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Section 8.2 Perimeter and Area of Polygons

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  1. Section 8.2 Perimeter and Area of Polygons The perimeter of a polygon is the sum of the lengths of all sides of the polygon. Table 8.1 and 8.2 p. 363 Ex 1,2 p. 364 Section 7.2 Nack

  2. Theorem 8.2.1Heron’s Formula for the Area of a Triangle • Semiperimeter s = ½ (a + b + c) • Heron’s Formula: A = s (s - a) (s - b) (s - c) • Example 3: Find the area of a triangle with sides 4, 13, 15. s = ½ (4 + 13 + 15) = 16 A = 16 (16 - 4) (16 - 13) (16 - 15) = 24 sq. units. Section 7.2 Nack

  3. Theorem 8.2.2: Brahmagupta’s Formula for the area of a cyclic* quadrilateral • Semiperimeter = ½(a + b + c + d) • Area = A =  (s - a) (s - b) (s - c) (s – d) *cyclic quadrilateral can be inscribed in a circle so that all 4 vertices lie on the circle. Section 7.2 Nack

  4. Area of a Trapezoid • Theorem 8.2.3: The area A of a trapezoid whose bases have lengths b1 and b2 and whose altitude has length h is given by: A = ½ h (b1 + b2 ) = ½ (b1 + b2)h The average of the bases times the height Proof p. 366 Example 4 p. 367 Section 7.2 Nack

  5. Quadrilaterals with Perpendicular Diagonals • Theorem 8.2.4: The area of any quadrilateral with perpendicular diagonals of lengths d1 and d2 is given by A = ½ ( d1d2 ). • Corollary 8.2.5: The area A of a rhombus whose diagonals of lengths d1 and d2 is given by A = ½ ( d1d2 ) • Corollary 8.26: The area A of a kite whose diagonals of lengths d1 and d2 is given by A = ½ ( d1d2 ) Proof: Draw lines parallel to the diagonals to create a rectangle. The area of the rectangle A = ( d1d2 ). Since the rectangle is twice the size of the kite, the area of the kite A = ½ ( d1d2 ) Ex. 6 p. 369 Section 7.2 Nack

  6. Areas of Similar Polygons • Theorem 8.2.7: The ratio of the areas of two similar triangles equals the square of the ratio of the lengths of any two corresponding sides: • Proof p. 369 • Note: This theorem can be extended to any pair of similar polygons (squares, quadrilaterals, etc.) Ex. 7 p. 370 Section 7.2 Nack

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