2.2+Nosing+Around

__ 2.2 Nosing Around  __  **   All the members of the Wump family have the same angle measures. Is having the same angle measures enough to make two figures similar? All rectangles have four right angles. Are all rectangles similar? What about these two rectangles? **  

**  These rectangles are not similar, because they don’t have the same shape-one is tall and skinny, and the other looks like a square. To be similar, it is not enough for figures to have the same angle measures. ** **   In this problem, you will investigate rectangles more closely to try to figure out what else is necessary for two rectangles to be similar. You will compare side lengths, angle measures, and perimeters. **  **   One way to compare two quantities is to form a // ratio //. **   **   For example, Mug Wump’s nose is 1 unit wide and 2   ** **  units long. To compare the width to the length, we can  ** **  use the ratio // 1 to 2, //which can also be written as the   ** **  fraction ½. **  ** __ Problem 2.2 __ ** ** Copy the chart below. The Wumps in the chart are numbered according to their size. Mug is Wump 1. Since the segment that makes up Zug are twice as long as the segments that makes up Mug, Zug is Wump two. Since the segments that make up Bug are three times longer than the segments that make up Mug, Bug is Wump 3. Since Lug and Thug are not similar to the others they are at the bottom of the chart. ** ** A)  **** Look carefully at the noses of Mug, Zug, Bug, Lug and Thug. In your table, record the dimensions, the ratio of width to length (Width/Length), and the perimeter of each nose.  **  **   B)   ****   Look at the data you recorded for Mug, Zug, Bug, Lug, and Thug. What patterns do you see? Explain how the values in each column change as the Wumps get bigger. Look for relationships between the values in the different columns. **  The width of the noses grow by 1, and the length is double the width. The width above length is always 1/2. **  C)   ****   The rule for making Wump 4 is (4x, 4y). The rule for making Wump 5 is (5x,5y). Add data to the chart for Wumps 4 and 5. Do their noses fit the patterns you noticed in part B?   **   Yes, their noses fit the patterns.    **   D)   ****   Use the patterns you found to add data for Wumps 10, 20, and 100 to the chart. Explain your reasoning. **  My reasoning is that the rule for making wup 10 is (10x,10y). So, you just follow the rule, and you fill in the table. **  E)   ****   Do Lug’s nose and Thug’s nose seem to fit the patterns you found for the Wumps? If not, what makes them different?   **   No, Lug and Thugs noses don’t fit the patterns. They have different widths and lengths for their noses. Also, both of them have different rules.   __  Problem 2.2 Follow-Up  __  **   1.   ****   Is there a scale factor from Mug’s nose to Wump 4’s nose? Why or why not?   **   Yes, there is because the side lengths and the perimeter grow by a scale factor of 4.   **   2.   ****   Is there a scale factor from Mug’s nose to Thug’s nose? Why or why not?   **   No, because you can’t multiply the side lengths of Mug’s nose to find the side lengths of Thug’s nose.   **   3.   ****   The dimensions of Bug’s nose are 3x6. Suppose this nose is enlarged by a scale factor of 3.   ** **   a.   ****   What are the dimensions of the new nose? **  The dimensions of the new nose are 9x18. **  b.   ****   What is the perimeter of the new nose? **  The perimeter of the new nose is 162. **  4.   ****   a. What is the scale factor from Wump 2 to Wump 10? **  The scale factor from Wump 2 to Wump 10 is 5. **  b. What is the scale factor from Wump 10 to Wump 2? **  The scale factor from Wump 10 to Wump 2 is 1/5.
 * ** Wump  ** || **  Width of Nose  ** || **  Length of Nose  ** || **  Width/ length  ** || **  Perimeter  ** ||
 * ** Wump 1  ** ||  1  ||  2  ||  ½  ||  6  ||
 * ** Wump 2  ** ||  2  ||  4  ||  2/4 = ½  ||  12  ||
 * ** Wump 3  ** ||  3  ||  6  ||  3/6 = ½  ||  18  ||
 * ** Wump 4  ** ||  4  ||  8  ||  4/8 =1/2  ||  24  ||
 * ** Wump 5  ** ||  5  ||  10  ||  5/10 = ½  ||  30  ||
 * ** Wump 10  ** ||  10  ||  20  ||  20/40 = ½  ||  60  ||
 * ** Wump 20  ** ||  20  ||  40  ||  20/40 = 1/2  ||  120  ||
 * ** Wump 100  ** ||  100  ||  200  ||  100/200 = ½  ||  600  ||
 * ** Lug  ** ||  3  ||  3  ||  1  ||  10  ||
 * ** Thug  ** ||  1  ||  7  ||  1/7  ||  15  ||