Math 53 "Winter ’09"
7.2 "Perimeter and Area of Polygons"
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Objectives:
*
Understand the idea of perimeter.
*
Learn how to use the Heron’s Formula and Brahmagupta’s Formula.
*
How to …nd the area of trapezoid, rhombus, and kite.
*
Use the ratio of the area of two similar triangles
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Preliminaries:
De…nition:
"Perimeter"
kThe perimeter of a polygon is the sum of the lengths of all sides of the polygonk
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It is important to understand the concept of perimeter rather than memorize formulas.
Perimeter of a Triangle
Scalene Triangle
Isosceles Triangle
Equilateral Triangle
Example 1: (Finding the perimeter of a triangle)
Find the perimeter of
ABC if the altitude AD = 8 cm; BC = 6 cm; and AB = AC
Page: 1
Bibiana Lopez
Elementary Geometry by Alexander and Koeberlein
7.2
Perimeter of a Quadrilateral
Quadrilateral
Rectangle
Square (or Rhombus)
Parallelogram
Example 2: (Finding the perimeter of a quadrilateral)
Find the perimeter of the following …gure
Heron’s Formula
If the lengths of the sides of a triangle are known, the formula used to calculate the area is Heron’s Formula.
Named in honor of Heron of Alexandria.
Theorem 7.2.1:
Heron’s Formula
If the three sides of a triangle have lengths a; b; and c, then the area A of the triangle
is given by
where the semiperimeter of the triangle is
Example 3: (Applying Heron’s Formula)
Find the area of a triangle whose sides measure are 10 cm; 17 cm; and 21 cm:
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Bibiana Lopez
Elementary Geometry by Alexander and Koeberlein
Theorem 7.2.2:
7.2
Brahmagupta’s Formula
For a cyclic quadrilateral with sides of lengths a; b; c; and d, the area
is given by
where
Example 4: (Applying Brahmagupta’s Formula)
For cyclic quadrilateral ABCD; …nd the area if AB = 6 cm; BC = 7 cm; CD = 2 cm; and DA = 9 cm:
Area of a Trapezoid
Theorem 7.2.3:
The area A of a trapezoid whose bases have lengths b1 and b2 and whose altitude has
length h is given by
Proof of Theorem 7.2.3
Given:
Trapezoid ABCD with AB
Prove:
AABCD = 12 h (b1 + b2 )
DC
Example 5: (Applying Theorem 7.2.3)
Find the area of the given polygon.
Page: 3
Bibiana Lopez
Elementary Geometry by Alexander and Koeberlein
7.2
Quadrilaterals with Perpendicular Diagonals
Theorem 7.2.4:
The area of any quadrilateral with perpendicular diagonals of lengths d1 and d2
is given by
Proof of Theorem 7.2.4
Given:
Quadrilateral ABCD with AC ? BD
Prove:
AABCD = 12 d1 d2
Example 6: (Finding the area of a kite)
Find the area of the kite ABCD
Areas of Similar Polygons
Theorem 7.2.7:
The ratio of the areas of two similar triangles equals the square of the ratio of the lengths
of any two corresponding sides; that is
Example 7: (Applying Theorem 7.2.7)
Find the ratio
A1
A2
of the areas of two similar rectangles if:
s1
s2
=
2
5
a)
The ratio of corresponding sides is
b)
The length of the …rst rectangle is 6 m; and the length of the second rectangle is 4 m:
Page: 4
Bibiana Lopez