Exercise 7 Page 574

We are asked to complete the graphic organizer to compare how the scale factor affects the side lengths, perimeters, and areas of similar rectangles.

Let's consider each of the given scale factors separately.

Scale Factor of 2

Let's consider an example rectangle.

Now let's find a rectangle that is similar by a scale factor of 2. To do so, we apply a dilation with a scale factor equal to 2 to the original rectangle.

First, let's note the lengths and widths of both rectangles.

The length of the rectangle after the dilation is 4, which is the length of the rectangle before the dilation, 2, multiplied by the scale factor 2. Likewise, the width of the rectangle after the dilation is 2, which is the width of the rectangle before the dilation, 1, multiplied by the scale factor 2. 2 * 2 = 4 1 * 2 = 2 Let's put these values into the graphic organizer.

Now we can consider the perimeters of the two figures. Let's recall the formula for the perimeter P of a rectangle with length l and width w. P = 2 l + 2 wLet's find the perimeter P_1 of the original rectangle. Here, the length l_1 is 2 and the width w_1 is 1.

The perimeter of the rectangle before the dilation is 6. Now let's find the perimeter P_2 of the rectangle after the dilation. This time, the length l_2 is 4 and the width w_2 is 2.

The perimeter of the rectangle after the dilation is 12. This is the same as the perimeter of the rectangle before the dilation, 6, multiplied by the scale factor, 2. 6 * 2 = 12 Let's put this information into the organizer.

Finally, let's consider the areas of the rectangles. Recall the formula for the area A of a rectangle with length l and width w. A = l * w We can find the area A_1 of the rectangle before the dilation by multiplying the length l_1= 2 and the width w= 1.

The area of the rectangle before the dilation is 2. For the rectangle after the dilation, the length l_2 is 4 and the width w_2 is 2.

The area of the rectangle after the dilation is 8, which is the same as the area of the rectangle before the dilation, 2, multiplied by the square of the scale factor 2. 2 * 2^2 = 8 This lets us complete the first row of the graphic organizer.

Now let's consider the situation where the scale factor is equal to 4.

Scale Factor of 4

Let's return to our original rectangle.

We already found that the length of this rectangle is 2, the width is 1, the perimeter is 6, and the area is 2. Now let's find a rectangle that is similar by a scale factor of 4 by applying a dilation of scale factor 4 to the original rectangle.

The length of the rectangle after the dilation is 8, the width is 4, and the perimeter is 24. Note that this is the same as the length, width, and perimeter, respectively, of the original rectangle multiplied by the scale factor 4.

The area of the rectangle after the dilation is 8* 4=32. This is the same as the area of the rectangle before the dilation, 2, multiplied by 16, the square of the scale factor.

Scale Factor of 0.5

Let's reset our original rectangle again.

The length of this rectangle is 2, the width is 1, the perimeter is 6, and the area is 2. Now let's apply a dilation with scale factor 0.5 to the original rectangle to find another similar rectangle.

The length of the rectangle after the dilation is 1, the width is 0.5, and the perimeter is 3. Note that this is the same as the length, width, and perimeter, respectively, of the original rectangle multiplied by the scale factor 0.5.

The area of the rectangle after the dilation is 0.5. This is the same as the area of the rectangle before the dilation, 2, multiplied by 0.25, or the square of the scale factor.

Scale Factor of 23

Let's consider the same starting rectangle again.

The length of this rectangle is 2, the width is 1, the perimeter is 6, and the area is 2. Now let's dilate the rectangle by a scale factor of 23.

The length of the rectangle after the dilation is 43, the width is 23, and the perimeter is 4. This is the same as the length, width, and perimeter, respectively, of the original rectangle multiplied by the scale factor 23.

The area of the rectangle after the dilation is 89, which is the same as the area of the rectangle before the dilation, 2, multiplied by 49, the square of the scale factor.

Scale Factor of k

In general, if two figures are similar by a scale factor k, then the length of a side of the image is k times greater than the length of the corresponding side of the preimage. The perimeter of the image is k times the perimeter of the preimage and the area of the image is k^2 times the area of the preimage.