Similarity Study Guide Name: _____________________________________ Class: _____________________ Date: ___________ State the three triangle similarity theorems: 1. ___________________________ 2. ____________________________ 3. _____________________________ Draw two triangles that demonstrate each of the similarity theorems (label as needed): 1. 2. 3. SCALE FACTOR (SF): If two shapes are similar, there will be a SCALE FACTOR that determines the relationship between the two shapes. This scale factor is defined as the value that, when multiplied by all aspects of a pre-image, can be used to determine all aspects of the image of that pre-image. In other words, the image divided by the pre-image gives you the scale factor! SF = 𝑰𝒎𝒂𝒈𝒆 𝑷𝒓𝒆−𝑰𝒎𝒂𝒈𝒆 Find the SCALE FACTOR between the following images: Pre-image PRQ is the pre-image SBR is the pre-image Pre-image MNL is the pre-image SER is the pre-image

Are these triangles similar Are these triangles similar? Use what you know about similarity theorems and scale factors to determine if the following triangles are similar. If they are, state the relevant similarity theorem. Using SCALE FACTOR to find the missing length. Determine the scale factor between each pair of triangles, and use that scale factor to find the missing side length. ASSUME THE SMALLER TRIANGLE IS THE PRE-IMAGE. SF _________ ? _________ SF _________ ? _________ 7.5 5 SF _________ ? _________ SF _________ GF is parallel to CD

Dilations and Scale Factor. Dilating on a coordinate plane is nothing more than applying a scale factor to the coordinates of each vertex of a shape. Therefore, the scale factor and the coordinate rule for dilations is THE SAME THING! Example – Dilating with a scale factor of 1.5 is written as: (x , y) → (1.5x , 1.5y) Try the following two dilations: What is the CENTER OF DILATION for the previous two problems? _____________________________ Figuring out the SCALE FACTOR by observing a dilation. When a pre-image is dilated to create an image, the two shapes will have a ________________ that can be multiplied by every aspect of the pre-image to discover every aspect of the image. Use this information to discover the scale factor between the following images on the coordinate plane (be careful identifying the pre-image and image!) X’ Y’ A X Y Z’ W’ A’ W Z SF = 𝑰𝒎𝒂𝒈𝒆 𝑷𝒓𝒆−𝑰𝒎𝒂𝒈𝒆 = SF = 𝑰𝒎𝒂𝒈𝒆 𝑷𝒓𝒆−𝑰𝒎𝒂𝒈𝒆 = SF = 𝑰𝒎𝒂𝒈𝒆 𝑷𝒓𝒆−𝑰𝒎𝒂𝒈𝒆 =

Similarity Transformations – Dilations combined with other transformations To prove that two shapes are similar on a coordinate plane, dilate the pre-image and translate/reflect/rotate until you map the pre-image onto the image. Apply similarity transformations to the following two pre-image/images. Apply the transformations to the necessary coordinate points, and draw each transformed shape as you go. A’’’ F(6,4) C’’’ B’’’ A B C F’’(-2,0) Step 1 – Dilate (SF = _______ ) (x , y) → ( ____ , ____ ) A( ___ , ___ ) → A’( ___ , ___ ) B( ___ , ___ ) → B’( ___ , ___ ) C( ___ , ___ ) → C’( ___ , ___ ) Step 2 – Reflect A’( ___ , ___ ) → A’’( ___ , ___ ) B’( ___ , ___ ) → B’’( ___ , ___ ) C’( ___ , ___ ) → C’’( ___ , ___ ) Step 3 – Translate (x , y) → ( ______ , ______ ) A’’( ___ , ___ ) → A’’’( ___ , ___ ) B’’( ___ , ___ ) → B’’’( ___ , ___ ) C’’( ___ , ___ ) → C’’’( ___ , ___ ) Step 1 – Dilate (SF = _______ ) (x , y) → ( ____ , ____ ) F( ___ , ___ ) → F’( ___ , ___ ) Step 2 – Translate (x , y) → ( ______ , ______ ) F’( ___ , ___ ) → F’’( ___ , ___ ) How did you determine the scale factor for the circle transformation? _______________________________________ Note – The order of transformations is not important as long as your pre-image is mapped onto your image. Dilated first is recommended, but not necessary!

Ratio of A : B = 𝑨 𝑨+𝑩 and 𝑩 𝑨+𝑩 Subdividing a segment by a ratio – understanding ratios in terms of fractions EXAMPLE: What does it mean to break something up by the ratio of 3 : 1? A ratio of 3:1 means that one piece of the object we are breaking up is 3 times the size of the other piece. 3 1 How many total pieces of the same size do you actually have? _____________ Thus, think of a ratio in terms of a fraction with the denominator being the number of pieces you have: Ratio of A : B = 𝑨 𝑨+𝑩 and 𝑩 𝑨+𝑩 So for the above ratio, your two pieces can be represented by two fractions: Ratio of 3 : 1 = and Convert the following ratios into a pair of fractions: 2 : 1 _________________________ 3 : 2 _____________________________ 4 : 1 _________________________ 5 : 3 _____________________________ 1 : 7 _________________________ 4 : 9 _____________________________ 1 : 1 _________________________ 12 : 3 ____________________________