7.2 Similar Polygons SOL: 14a Objectives: Identify similar figures. Solve problems involving scale factors.

Similar Figures The same shape but not necessarily the same size Examples:

Congruent VS Similar

Similar Polygons Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional (same shape, different size) Symbol ~ is read is similar to Example: ABCD ~ EFGH

Similar Polygons Order of the vertices in a similarity statement is important Similarity Statement ABCD ~ EFGH  D   H D   H DAHEDAHE It identifies the corresponding angles and the corresponding sides Congruent AnglesCorresponding Sides  B   F B   F  C   G C   G  A   E ABEFABEF = BCFGBCFG = CDGHCDGH =

Examples 1: Determine whether each pair of figures is similar. Justify your answer.  L = 70 ,  M = 70   R = 65 ,  Q = 65  Since the corresponding angles are not congruent then the triangles are not similar 180  - 40  = 180  - 50  = 140  ÷ 2 = 70  130  ÷ 2 = 65 

Determine whether each pair of figures is similar. Justify your answer. Example 2: CA TR ? = Since the three ratios are the same, then the triangles are similar So we can say, ΔABC ~ ΔRST 1 st Check for equal ratios of the corresponding sides to see if they are similar triangles: 1.333… = ? BC ST AB RS ? = ? = 8686 = 1.333… 60 40

Scale Factor Usually used for models of real-life objects Numerical ratio, comparing corresponding sides of similar figures Examples: - model cars - models of architecture - special effects in movies - maps

Example 3: An architecture prepares a 12-inch model of a skyscraper to look like a real 1100-foot building. What is the scale factor of the model compared to the real building? The real-life object is the skyscraper, which is measured in feet. The model is in inches, so you need to convert the model measurement into feet. 12-inches = 1 foot, So our scale factor would be

Example 4: The two polygons are similar. Write a similarity statement. Then find x, y, and UV. UV = y + 1 = = y + 1 = ABCDE ~ RSTUV 6(3) = 18 = 4x 18 4x = = x 6(y + 1) = 6y + 6(1) = 32 6y + 6 = y = 26 6y 26 6 = y = = == x x = 4(8)

Example 5: Find the scale factor of polygon ABCDE to polygon RSTUV = :4 =

Example 6: Rectangle WXYZ is similar to rectangle PQRS with a scale factor of 1.5. If the length and width of PQRS are 10 meters and 4 meters, respectively, what are the length and width of rectangle WXYZ? l W ZY X PQ RS 10 4 w l 10 = 3 2  WXYZ  PQRS 2l = 2l = 30 10(3) l = 15 w 4 = 3 2  WXYZ  PQRS 2w = 2w = 12 4(3) w = 6

Example 7: The scale on the map of a city is ¼ inch equals 2 miles. On the map, the width of the city at its widest point is 3 ¾ inches. The city hosts a bicycle race across town at its widest point. Tom bikes at 10 miles per hour. How long will it take him to complete the race? How long will it take Tom to bike 30 miles? = ¼ 3 ¾ 2 x 0.25x = 7.5 1st how many total miles: x = 30 miles 30 / 10 = x 0.25x = = 3 hrs x = (3.75)(2)