Introduction to Similar Figures
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- Allyson Martin
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1 Introduction to Similar Figures In this section we will learn how to determine whether two polygons are similar to each other. Next, note that the order of the letters is important in ABC ~ DEF.. This is telling us that A D, in this case A D 49 B E, in this case B E 109 C F, in this case C F 22 This has demonstrated that the corresponding angles are congruent. For two polygons to be similar, their corresponding angles must be congruent (the same) and their corresponding sides must be proportional. Let s show how ABC ~ DEF. Next, we use the order of the letters again to determine the corresponding sides. mde mef mca mfd First, note that the symbol ~ means that two polygons are similar It is now confirmed that ABC ~ DEF. 1
2 It is important to note that for triangles only one of the conditions needs to be met; that is, for triangles, their corresponding angles must be congruent or their corresponding sides must be proportional. If we know that two polygons are similar, we can find unknown measurements in the given polygons. Example 1. From the triangles given below, determine which two triangles are similar to each other. For example, determine the missing measurement x in the rectangle below where ABCD ~ EFGH First, we identify which sides are proportional to each other. mef mfg 4 7 x 10.5 ( 4)( 10.5) ( x)( 7) 42 ( x)( 7) 42 Step 1: Can we add information to any of the three triangles? Since all 3 sides of DEF. measure 5 units, this is an equilateral triangle. This means that all 3 interior angles measure 60 each. 7 x x 6 We are ready to do more challenging examples. 2
3 Step 2: Determine which two triangles are definitely similar to each other. Step 3: Recall that the interior angles of a triangle add up to 180 to solve for x. D+ E+ F x x 180 x x 60 ABC ~ DEF because their corresponding angles are congruent. Note that there was not enough information about GHI to determine whether or not it was similar to the other two triangles. 2. Determine the missing angle x in the triangle below where ABC ~ DEF. 3. Determine the missing measurement x in the rectangle below where ABCD ~ EFGH. Step 1: Match up the congruent corresponding angles. A D, A 50 so D 50 B E, E 70 so B 70 C F Step 2: Label the triangles with the new information from Step 1. Step 1: Determine which sides are proportional to each other. mef mfg 3
4 Step 2: Input the values and solve for x. mef mfg 6 x ( 6)( 14.4) ( 5)( x) 86.4 ( 5)( x) 86.4 Step 2: Multiply the dimensions of Oliver s sandbox by 2 to get the dimensions of Amanda s sandbox. Oliver ( 6)( 2) 12 ( 8)( 2) 16 Amanda 5 x x Amanda and Oliver each have their own rectangular sandbox. The perimeter of Amanda s sandbox is twice as big as the perimeter of Oliver s. Determine the dimensions of Amanda s sandbox if the dimensions of Oliver s sandbox are 8 feet by 6 feet and the sandboxes are similar. The dimensions of Amanda s sandbox are 12 by 16. Step 1: Draw Oliver s sandbox and label its sides. 4
5 5. Determine whether two squares are always similar and explain your answer. Step 1: Draw two squares and determine if they are similar. The corresponding sides of the two squares are proportional no matter what their measures are. This proves that two squares are always similar. Is ABCD ~ EFGH? mcd mda mef mfg mgh mhe The corresponding sides are proportional so ABCD ~ EFGH. Step 2: Prove that the result from Step 1 is true for all squares. mcd mda mef mfg mgh mhe x x x x y y y y 5
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