Warm-Up 5 minutes IF you have an appropriate device, find at least one example of where ratios and proportions are used in the real world Think about it Before a building is built, an architect has to first build a model of what they want the building to look like. How do you think that their design is transformed into the actual building? Fashion? Theater? Music? Food? Manufacturing? Architecture? Medicine?
homework
Chapter 6: Similarity 6.3: Using Similar Polygons Objectives: If two figures are similar, how do you find the length of a missing side?
Terms: Ratio: Written as either x y or x : y Scale Factor: Ratio of the lengths for 2 corresponding sides. To find the scale factor, what shape you are going TO over where you are coming FROM: TO Similar/Similarity: FROM Similar figures are figures that have the same shape but not necessarily the same size.
Similar Figures Similar Figures ( ) mean similar to Two figures are similar if two conditions are true: 1. Corresponding ANGLES are congruent 2. Corresponding SIDES are proportional After proving triangles are similar, we can prove that their parts are congruent, too! CASTC: Corresponding Angles of Similar Triangles are Congruent CSSTP: Corresponding Sides of Similar Triangles are Proportional
Similarity V Example: Pentagon PQRST VWXYZ Z W List all the corresponding parts of the pentagon that are similar: Y X Angles: <P <V <Q <W Sides: PQ QR VW WX <R <X RS XY <S <Y ST YZ <T <Z PT VZ *When you name similar figures, be sure to name corresponding vertices IN THE SAME ORDER!!!
How to put vertices in order You are given the following: 5/28/2013 Free template from www.brainybetty.com The sides are going to match up so that the small sides are proportional, the middle sides are proportional, and the big sides are proportional. 1. Box the smallest number of each triangle with blue. 2. Circle the largest number of each triangle with green. 3. Triangle the number left over in red. 4. Now you can match up your vertices using color. Given CAT, you know that since C is between blue and red, it will have to match up with the vertex between blue and red on the other triangle, or G.
Scale Factor If 2 polygons are similar, then the ratio of the lengths of two corresponding sides is the scale factor of the similarity. #1 #2 8 To find the scale factor of figure #1 to figure #2: TO FROM 2 #2 #1 2 8 So, the scale factor is 4 1 If the figure gets smaller, then SF <1 If the figure gets larger, then SF >1 If the figure stays the same, then SF = 1 1 4
Example Quad ABCD Quad EFGH. Find the 1)Scale Factor to ABCD and (2) values of x, y, and z. #1 #2 Scale Factor: #1 #2 : 50 30 5 3 Value of x, y, and z: #1 #2 5 50 : # 1 5 : y #1 5 : 3 x #2 3 12 #2 3 x = 30 y = 20 z = 13.2 22 z
Always, Sometimes, Never For each statement, determine whether the statement is always (AT), sometimes (ST), or never (NT) true: a. Two rectangles are similar. b. Two squares are similar. AT ST c. A triangle is similar to a quadrilateral. d. Two isosceles triangles are similar. e. Two equilateral triangles are similar. ST AT NT
1. Get a partner Finding Sides of Similar Figures Directions 2. Get out a sheet of paper between the two of you. Fold the paper like a hot dog, and write each person s name at the top of a column. 3. Student B will copy the problem on his side of the paper and work the problem while Student A watches. 4. Repeat Step 2, reversing roles until all the problems are finished. 5. Turn in, after checking answers...
Finding Sides of Similar Figures Activity For the following problems, the figures are similar. Find the value of each variable. The figures are not drawn to scale! 1. 2. 35/9 11.2 5.4 7.2 3. 4. 20 6 4.5 14 10.5
Finding Perimeters of Similar Figures If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.
Mario s Pizzeria needs to rent a bigger space. He currently has a rectangular restaurant that is 60 feet long and 40 feet wide. He would like to keep the same shape of his restaurant but expand it so that it will be 85 feet long. What is the scale factor of the old restaurant to the new one he wants? Finding Perimeters of Similar Figures- Example TO FROM 85 60 What will be the perimeter of the new restaurant? x = 283.333 ft 17 12
Chapter 6: Similarity 6.4 and 6.5: AA Similarity Postulate and the SSS and SAS Similarity Theorems Objectives: How do you prove triangles similar?
A O C T D G 1. What appears to be true about the triangles above? angles? sides? s same size? s same shape? Similarity statement: ~ C T 5/28/2013 Free template from www.brainybetty.com
AA Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are. similar
AA Similarity Example 1. What do you know about right angles? 2. You also know that vertical angles are congruent! 3. The triangles are similar by AA Similarity.BUT what is the order of the vertices? ΔVWX ~ ΔZYX
B 24 9 E A 16 C D 6 F 2. What appears to be true about the following triangles? angles? sides? s same size? s same shape? Similarity statement: ~ A AB DE Free template from www.brainybetty.com
SAS Similarity Theorem SAS: If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. 5/28/2013 Free template from www.brainybetty.com
B 6 E 9 16 24 C 7.5 F A 20 D 3. What appears to be true about the triangles? angles? sides? s same size? s same shape? Similarity statement: ~ Compare smallest, largest... 5/28/2013 Free template from www.brainybetty.com
SSS Similarity Theorem SSS: If three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar. 5/28/2013 Free template from www.brainybetty.com
SSS Similarity Theorem Example AB EF CA DE CB DF AB EF CA DE CB DF 4 2 6 3 6 3 2 2 2 Since the sides all have the same scale factor, they are proportional.
SAS Similarity Theorem Example 1. The only way these could be similar is by SAS Similarity Theorem. 2. The small sides of each triangle have to be proportional, and the big sides of each triangle have to be proportional so that: RS RT XY XZ 3. Fill in the numerical values to make sure the sides are proportional. RS 4.2 RT 5.7 3 3 XY 1.4 XZ 1.9
1. Draw a picture and label the information. 2. Make sure the triangles are similar if you have parallel lines, look for Alternate. interior angles are congruent
3. Because you know the triangles are similar by AA, you can set up your proportions and solve for x. AB AE DC DE 4 (3x 4) 8 ( x 12) 4( x 12) 8(3x 4) 16 x 20 4 x 5 4. Now that you know x, you can find AE and DE: AE = DE = 4 2 3( ) 4 6 5 5 4 4 12 12 5 5
Similar Figures Proof Given: <B <C Prove: BM NM MC ML Example Statements Reasons 1. <B <C 1. given 2. <1 <2 2. Vertical angles are congruent 3. Δ BNM ~ΔCLM 3. AA Similarity Postulate 4. BM MC NM ML 4. Corresponding Sides of Similar Triangles are Proportional (CSSTP)
Class Examples X meters To estimate the height of a tree, a girl scout sights the top of the tree in a mirror that is 34.5 meters from the tree. The mirror is on the ground and faces upward. The scout is 0.75 meters from the mirror, and distance from her eyes to the ground is about 1.75 meters. How tall is the tree? 1.75 meters 34.5 meters 0.75 meters 5/28/2013 Free template from www.brainybetty.com 1.75/x =.75/34.5 60.375 =.75x 80.5 meters= x