Transcript Section 8.2 Perimeter and Area of Polygons perimeter

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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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 (4 + 13 + 15) = 16
A = 16 (16 - 4) (16 - 13) (16 - 15) = 24 sq. units.
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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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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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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 ( d1 d2 )
Ex. 6 p. 369
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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:
A1  a1 
  
A2  a2 
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• 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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