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Transcript

Similar Triangles Tutorial 12g Similarity When people say that two things are similar, usually they mean that the things are alike. In geometry, similar figures are alike in a very specific way. Figures that are similar have the same shape but may or may not have the same size. Here are three photos: the original photo, the enlargement of the original, and a reduction of the original photo. Because all three photos show the same shape, the figures are similar. Similar Polygons Two polygons are in correspondence when consecutive sides and vertices of one are matched with consecutive sides and vertices of the other. Similar Polygons For two polygons to be similar they must have: 1. Corresponding (or matching) angles that have the same measure. In the two rectangles below, all four angles in each are right angles, therefore they meet the first criteria of corresponding angles having the same measure. 2. Corresponding (or matching) sides that are proportional. Are all corresponding sides proportional? 8 12 Does 10 15 ? Yes, these ratios are equal. Therefore these two rectangles also meet the second criteria of corresponding sides being proportional. 10 in. 8 in. 15 15 in. 12 in. The ratios of the lengths of corresponding sides are equal. Similar Triangles Triangles have some special relationships between sides and angles when they are similar. It can be very helpful in mathematics to determine when two triangles are similar. If two triangles are similar, it becomes easier to learn more characteristics of the two triangles. It is not always necessary to prove both criteria, that all 3 corresponding angles have equal measures and all corresponding sides are proportional to determine whether or not two triangles are similar. The following mathematical rules give us two short cuts: Similar Triangles cont. . . Angle Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. D A Since A D & B E, then ABC is similar to DEF 43 43 B C E F “” means “is similar to”. So we can write the similarity using symbols: ABC DEF. Similar Triangles cont. . . Angle Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. D A Since A D & B E, then ABC is similar to DEF 43 43 B C E F “” means “is similar to”. So we can write the similarity using symbols: ABC DEF. Similar Triangles cont. . . Since these two triangles are similar, we can use proportions to find the length of the missing side. D Since the two triangles are similar, the ratios of the corresponding sides are proportional. Therefore: 6 8 x 12 So, 8x = 6•12 8x = 72 x=9 Lets check our answer: 6 8 Does ? YES! 9 12 43 A 43 12 in. 8 in. B 6 in. C E x F Similar Triangles cont. . . SSS Similarity Theorem If the sides of two triangles are in proportion, then the triangles are similar. Example: POM LMN, what are the lengths of the missing sides? L 7 Solution: ? 10.5 ? O 7 PM = 7 PO = 6 LN = 10.5 MN = 12 LM = ? MO = ? 7 12 M 6 P 10 .5 N 6 10 .5 LM 7(LM) = 10.5 6 7(LM) = 63 7 ( LM ) 63 7 7 LM = 9 6 LM MO 12 7 10 .5 MO 12 10.5(MO) = 12 7 10.5(MO) = 84 10 .5( MO ) 84 10 .5 10 .5 MO = 8