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
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2. Theorem 8.2.1 Heron’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 ( ) = 16
A = 16 (16 - 4) ( ) ( ) = 24 sq. units.
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3. Theorem 8. 2. 2: Brahmagupta’s Formula for the area of a cyclic
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.
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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
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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
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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
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