The Scaling Principle for Area

New York State Common Core Math Geometry, Module 3, Lesson 3

Worksheets for Geometry, Module 3, Lesson 3

Student Outcomes

• Students understand that a similarity transformation with scale factor r multiplies the area of a planar region by a factor of r².
• Students understand that if a planar region is scaled by factors of a and b in two perpendicular directions, then its area is multiplied by a factor of ab.

The Scaling Principle for Area

Classwork

Exploratory Challenge

Complete parts (i)–(iii) of the table for each of the figures in questions (a)–(d): (i) Determine the area of the figure (preimage), (ii) determine the scaled dimensions of the figure based on the provided scale factor, and (iii) determine the area of the dilated figure. Then, answer the question that follows. In the final column of the table, find the value of the ratio of the area of the similar figure to the area of the original figure. e. Make a conjecture about the relationship between the areas of the original figure and the similar figure with respect to the scale factor between the figures.

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Exercises 1–2

1. Rectangles 𝐴 and 𝐵 are similar and are drawn to scale. If the area of rectangle 𝐴 is 88 mm², what is the area of rectangle 𝐵?
2. Figures 𝐸 and 𝐹 are similar and are drawn to scale. If the area of figure 𝐸 is 120 mm², what is the area of figure 𝐹?

Lesson Summary

THE SCALING PRINCIPLE FOR TRIANGLES: If similar triangles 𝑆 and 𝑇 are related by a scale factor of 𝑟, then the respective areas are related by a factor of 𝑟².

THE SCALING PRINCIPLE FOR POLYGONS: If similar polygons 𝑃 and 𝑄 are related by a scale factor of 𝑟, then their respective areas are related by a factor of 𝑟².

THE SCALING PRINCIPLE FOR AREA: If similar figures 𝐴 and 𝐵 are related by a scale factor of 𝑟, then their respective areas are related by a factor of 𝑟².

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