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What happens to the area of a figure if we scale it up or down (i.e., enlarge or reduce it)? In Part C, we review the concept of similarity and examine the relationship between a scale factor and the resulting area of the similar figure. Previously we only explored similar triangles, but in this section we will use a variety of shapes.
When we enlarge or reduce a figure, we are using an important mathematical idea: similarity. Similar figures have the same shape but are not necessarily the same size. More formally, we state that two figures are similar if and only if two things are true: (1) The corresponding angles have the same measure, and (2) the corresponding segments are in proportion. Enlarging or reducing a figure produces two figures that are similar. Note 3
The second attribute means that when we are building a similar figure, we must increase or decrease the sides multiplicatively by the scale factor. What happens to the length of each side when we enlarge a figure, say, by a scale factor of 2? Well, since in similar figures the corresponding sides are in proportion, each of the sides of the enlarged similar figure is twice as long as the corresponding side of the original figure. So, for example, in the enlargement of the trapezoid shown below on the left, the enlarged trapezoid is similar to the small trapezoid because the angles are congruent and each of the sides is proportionally larger (twice as long):
Building similar figures, however, is not always so straightforward! For example, the trapezoid below is not similar to the original trapezoid. The angles are congruent, but the corresponding sides are not proportional — some of the sides have been “stretched” more than others:
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.)
Problem C1
Problem C2
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.
Polygon |
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 |
Rectangle C | |||
Parallelo- gram M |
|||
Trapezoid K |
Problem C3
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?
Problem C4
What is the relationship between the scale factor and the area of the enlarged figure?
Problem C5
If the area of a polygon were 8 cm^{2}, 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 cm^{2} and the enlargement of that rectangle using a scale factor of 3.
Problem C6
If the scale factor of an enlargement is k, explain why the enlarged area is k^{2} times greater than the original area.
Problem C7
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?
Note 3
You can further explore the notion of similarity in Learning Math: Geometry, Session 8.
Note 4
If you are working in a group, you can divide the tasks in this problem. You may also want to try scaling up Polygons B, F, H, and L.
In this problem, you use the polygons to create similar figures. You may use more than one type of shape to build the similar figures, but you do not have to. Be sure that the figures you create satisfy both of the requirements for similarity — you may need to measure the sides and angles of the figures to check.
Polygon H, the hexagon, poses a challenge. You can use hexagons, trapezoids, and equilateral triangles to build the similar hexagons in this problem.
Problem C1
Problem C2
If A_{o} is the area of the original polygon, then we can write the following:
Polygon |
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 |
Rectangle C | 4 • A_{o} | 9 • A_{o} | 16 • A_{o} |
Parallelo- gram M |
4 • A_{o} | 9 • A_{o} | 16 • A_{o} |
Trapezoid K | 4 • A_{o} | 9 • A_{o} | 16 • A_{o} |
Problem C3
The number of copies needed is the square of the scale factor. For example, making a copy that is three times larger in each direction will take nine copies of the original shape.
Problem C4
The area of the enlarged figure is the original area multiplied by the square of the scale factor.
Problem C5
Because the scale factor is 3, the area is nine times larger. Therefore, the area of the enlarged figure is 72 cm^{2}.
For example, suppose the original figure were a 4-by-2 rectangle (with an area of 8 cm^{2}). The new shape would then be 12 by 6, with an area of 72 cm^{2} — nine times the original area. Here’s how it breaks down:
A = 12 • 6
A = (4 • 3) • (2 • 3)
A = 4 • (3 • 2) • 3associative property
A = 4 • (2 • 3) • 3commutative property
A = (4 • 2) • (3 • 3)associative property
Problem C6
One way to think about it is that enlarging an object will require k copies of that object in each direction: k copies in one direction, multiplied by k copies in the other direction, for a total of k^{2}.
Problem C7
All of these polygons are rep-tiles. Most rep-tiles have side lengths that have a common factor, but this is not a requirement.