Print several copies of the sample polygons (PDF) to use in the problems that follow. You'll need to cut out each of the polygons in order to manipulate them. (If you have access to Power Polygons, you can use them for these problems.)
In this problem, you will build similar figures by enlarging Triangle N. Use multiple copies of Triangle N to build the enlarged triangles. First use a scale factor of 2, then a scale factor of 3, and then a scale factor of 4. Check to make sure that all corresponding sides are proportionally larger than the original polygon. Sketch your enlargements.
What happens to the area of a figure that you enlarge by a scale factor of 2? Is the area of the enlarged figure twice that of the original?
Use Rectangle C, Parallelogram M, and Trapezoid K to build similar figures with scale factors of 2, 3, and 4, respectively. Then calculate the area of each enlargement in terms of the original polygon, and record your results in the table below. Note 4
For example, it takes four copies of the original trapezoid to make a similar shape with a scale factor of 2, so the enlarged trapezoid has an area that is four times greater than the original:
You do not have to build the enlargements of each shape using only that type of polygon, but you will need to determine the area of each enlargement in terms of the original polygon.
If Ao is the area of the original polygon, then we can write:
Scale Factor of 2 -- Area of the Enlargement in Terms of the Original Shape
Scale Factor of 3 -- Area of the Enlargement in Terms of the Original Shape
Scale Factor of 4 -- Area of the Enlargement in Terms of the Original Shape
Examine your enlargements. What is the relationship between the scale factor and the number of copies of the original shape needed to make a larger similar shape?
What is the relationship between the scale factor and the area of the enlarged figure?
If the area of a polygon were 8 cm2, what would the area of an enlargement with a scale factor of 3 be?
You may want to sketch a rectangle that has an area of 8 cm2 and the enlargement of that rectangle using a scale factor of 3. Close Tip
If the scale factor of an enlargement is k, explain why the enlarged area is k2 times greater than the original area.
Video Segment If you increase the size of a shape by a certain scale factor, what does that do to its area? Watch this segment to see how Michelle and David use different polygons to explore this question.
What happens to the area of a shape if you decrease its size by a certain scale factor? Does the same rule apply?
If you are using a VCR, you can find this segment on the session video approximately 16 minutes and 24 seconds after the Annenberg Media logo.
A rep-tile is a shape whose copies can be put together to make a larger similar shape. Look at Polygons B, C, F, L, and M. Which of these are rep-tiles? What do you notice about all of the rep-tile shapes?