Lesson 11.7 - Comparing Areas of Similar Figures

Overview of Area Ratios in Similar Figures

  • Focus on comparing areas of different shapes, particularly triangles and polygons.

Key Concepts

  • Area of Triangle:

    • Formula: A=12×base×heightA = \frac{1}{2} \times \text{base} \times \text{height}
    • Use base and height measurements to calculate area.

  • Similarity in Triangles:

    • Triangles are similar if all three pairs of corresponding angles are congruent (Angle-Angle similarity).
    • If triangles are similar, the ratio of their corresponding sides is constant.

  • Ratio of Sides:

    • For two similar triangles with sides of length 4 and 6, the ratio is:
      Sides Ratio=46=23\text{Sides Ratio} = \frac{4}{6} = \frac{2}{3}
    • This ratio holds true for all corresponding linear measurements (including heights and bases).

  • Ratio of Areas:

    • The ratio of areas of similar triangles is the square of the ratio of their corresponding sides:
      Area Ratio=(Side 1Side 2)2\text{Area Ratio} = \left(\frac{\text{Side 1}}{\text{Side 2}}\right)^2
    • Example with sides 2x and 3x for two triangles yields an area ratio:
      (2x)⋅(2h)(3x)⋅(3h)=49\frac{(2x)\cdot(2h)}{(3x)\cdot(3h)} = \frac{4}{9}

  • Theorem for Similar Figures:

    • If two figures are similar:
      Area 1Area 2=(Side 1Side 2)2\frac{\text{Area 1}}{\text{Area 2}} = \left(\frac{\text{Side 1}}{\text{Side 2}}\right)^2
    • This theorem applies to all types of similar figures, including polygons, not just triangles.

Practical Examples

  • Example with Pentagons:

    • Given two similar pentagons with side lengths 12 and 9, the area ratio calculation is as follows:
    • Ratio of sides: 129=43\frac{12}{9} =\frac{4}{3}
    • Area ratio becomes: (43)2=169\left(\frac{4}{3}\right)^2 = \frac{16}{9}

  • Example with Parallelograms:

    • Given areas of two similar parallelograms as 49 and 121, to find the side ratio:
    • Area ratio: 49121=(Side 1Side 2)2\frac{49}{121} = \left(\frac{\text{Side 1}}{\text{Side 2}}\right)^2
    • Solving yields: Side 1Side 2=711\frac{\text{Side 1}}{\text{Side 2}} = \frac{7}{11}

Additional Scenarios

  • Comparing Non-Similar Triangles:
    • If two triangles are not confirmed similar, calculate area using base and height relations and utilize the area ratio:
    • If heights and bases are derived directly from given properties, ratios can still be established based on area formulas.

Conclusion

  • Understanding the relationship between similar shapes and their areas, with an emphasis on the ratios, is crucial.
  • This knowledge can be applied to any similar figures to derive dimensions and areas accurately.