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 = \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:
    \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:
    \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:
    \frac{(2x)\cdot(2h)}{(3x)\cdot(3h)} = \frac{4}{9}

• Theorem for Similar Figures:

  • If two figures are similar:
    \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: \frac{12}{9} =\frac{4}{3}
  • Area ratio becomes: \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: \frac{49}{121} = \left(\frac{\text{Side 1}}{\text{Side 2}}\right)^2
  • Solving yields: \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.