Ratios of Areas 11'7

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Ratios of Areas 11.7
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Areas of Similar Polygons
GCP_Notes\area proportionality.gsp
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Areas of Similar Polygons

• Theorem
• If two polygons are similar with corresponding
  sides in the ratio of ab, then the ratio of the
  areas is a2b2

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If two polygons are similar, the ratio of the
perimeters of the polygons is equal to the ratio
of the corresponding sides.
p1326 p23618 (6/2)(18/6)
6
2
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If two polygons are similar, the ratio of the
areas of the polygons is equal to the square of
the ratio of the corresponding sides.
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2
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• The figures shown below are similar. Find the
  ratio of their perimeters and the ratio of their
  areas

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p21 A41
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6
8
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• What is the effect on the circumference of a
  circle if its radius is tripled?
• What happens to the area?

PnewPold31
AnewAold91
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• An architect draws plans for the construction of
  a new rectangular garage. The plans are drawn to
  the scale of ¼ inch to 1 foot, or 148. What is
  the ratio of the areas of the drawing and the
  actual garage?

Because the ratio of the dimensions of the
rectangles is 148, the ratio of the areas is
12482 or 12304.
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• In the figure below, find the ratio of the areas
  of the inscribed and circumscribed equilateral
  triangles ?X Y Z and ?P Q R. The length of CY is
  6 inches.

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• Find the length of the side of each equilateral
  triangle using 30- 60- 90 triangles. In CWY,
  YW 3?(3) therefore YZ 6?(3). In CPY, PY
  6?(3) therefore PR 12?(3).

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• Using the Areas of Similar Polygons Theorem, two
  equilateral triangles are similar polygons with
  corresponding sides in the ratio of 6?(3) 12?(3)
  or 12. Then the ratio of the areas is 12 22
  or 14.

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11.10.d
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11.7.21Find the ratio of areas
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20p64
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Polygon Equations

• Sum of exterior angles 360
• One exterior angle 360/n
• Sum of interior angles (n-2)180
• One interior angle of a regular polygon
• ? (n-2)180/n

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• Given r find a
• Cos ? a/r
• arcos ?
• Given a find r
• Cos ? a/r
• ra/cos ?
• Given r find s
• Sin ? s/2/r
• s2rsin ?
• Given a find s
• Tan ? s/2/a
• s2atan ?

?
a
r
s/2
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Regular Polygon Equations
Parallelogram Trapezoid Hexagon Reg. Poly
Area Reg. Poly Perimeter

• Triangle
• Square
• Rectangle
• Rhombus or
• Kite

Regular Polygon Areas
Regular Polygon Perimeters
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Regular Polygon Perimeters
Perimeter
Side Apothem Radius
Pentagon PerimeterPentagon
Regular Polygon Equations
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Regular Polygon Areas

• Area given the following measures

Side Apothem Radius
Regular Polygon Equations
Pentagon