Topic 3 Scale Factors and Areas of 2-D Shapes Unit 8 Topic 3

Explore Any shape can be enlarged or reduced by multiplying each of its dimensions by the same linear scale factor. 1.Work with a partner to complete the table below. 2. What is the relationship between the linear scale factor and the area scale factor? Try this on your own first!!!!

Explore What is the relationship between the linear scale factor and the area scale factor? Try this on your own first!!!! You should notice that the area scale factor is equal to the linear scale factor squared

Information The relationship between the area of a new shape and the area the original shape can be expressed using the following equation. Area Scale Equation new area = old area k 2 where k is the linear scale. We can rearrange the equation to isolate the area scale factor, k 2. Area Scale Factor (ASF)

Example 1 Determining a New Area Maggie scanned an 8” by 10” photograph of a hummingbird to her computer so that she could change the size. a) If the photograph is enlarged by a linear scale factor of 4, then determine the area of the enlarged photograph. Method 1: Using Area Calculation Find the area of the enlarged picture. Try this on your own first!!!!

Example 1a: Solution Determining a New Area Method 1: Using Area Calculation Find the area of the enlarged picture.

Example 1a: Solution Determining a New Area Method 2: Using the Area Scale Equation Substitute into the area scale equation

Example 1b: Solution Determining a New Area b) Suppose Maggie decided to decrease the size of the original photograph by a linear scale factor of. What is the area of the reduced image?

Example 1c: Solution Determining the Area Scale Factor c) Determine the area scale factor if the linear scale factor is.

Example 2 Determining the scale factor of an enlargement Try this on your own first!!!! Jim’s laptop has a monitor with the dimensions 9 in by 12 in. The image on his laptop is projected onto a screen. The image on the screen, which is similar to that on the laptop, has an area of in 2. By what factor did the area of the screen increase by? (That is, how many times greater is the area of the screen than the laptop?)

Example 2a: Solution Jim’s laptop has a monitor with the dimensions 9 in by 12 in. The image on the screen, which is similar to that on the laptop, has an area of in 2. By what factor did the area of the screen increase by?

Example 2b: Solution Determining the scale factor of an enlargement Determine the linear scale factor used to project the image from the laptop to the screen.

Example 3 Determining the Area Given a Scale Diagram Mr. and Mrs. Smith recently moved into a new home. In their rectangular backyard, they have a rectangular patio and a circular koi fish pond in the corner, as shown in the scale diagram below. a) If the radius of the pond in the diagram is 1.5 cm, what is the area of the diagram koi pond, to the nearest tenth? Try this on your own first!!!!

Example 3a: Solution Determining the Area Given a Scale Diagram Mr. and Mrs. Smith recently moved into a new home. In their rectangular backyard, they have a rectangular patio and a circular koi fish pond in the corner, as shown in the scale diagram below. a) If the radius of the pond in the diagram is 1.5 cm, what is the area of the diagram koi pond, to the nearest tenth?

Example 3b: Solution Determining the Area Given a Scale Diagram Determine the area, to the nearest tenth of a square cm, of the actual pond if the diagram was drawn using a linear scale factor of 0.01.

Need to Know: The area scale factor, ASF, of a 2D shape is The area of the original or old shape is multiplied by the area scale factor to produce the area of the new shape. The area scale equation is new area = old area k 2, where k is the linear scale factor. You’re ready! Try the homework from this section.