NOTES 8.2 Similarity

Definition: Similar polygons are polygons in which: The ratios of the measures of corresponding sides are equal. Corresponding angles are congruent.

Similar figures: figures that have the same shape but not necessarily the same size. Dilation: when a figure is enlarged to be similar to another figure. Reduction: when a figure is made smaller it also produces similar figures.

Proving shapes similar: Similar shapes will have the ratio of all corresponding sides equal. Similar shapes will have all pairs of corresponding angles congruent.

Example: ∆ABC ~ ∆DEF = = = 8 12 4 6 B C E F 5 10 Therefore: A corresponds to D, B corresponds to E, and C corresponds to F. The ratios of the measures of all pairs of corresponding sides are equal. = = =

Each pair of corresponding angles are congruent. <B <E <A <D <C <F

∆MCN is a dilation of ∆MED, with an enlargement ratio of 2:1 for each pair of corresponding sides. Find the lengths of the sides of ∆MCN. C (0,8) 8 MC = MN = CN = E (0,4) 6 10 D N M (0,0) (3,0) (6,0)

Given: ABCD ~ EFGH, with measures shown. 1. Find FG, GH, and EH. FG = GH = EH = 6 B 6 F 9 4 A 4.5 A E C 7 3 D G 10.5 H PABCD = 20 PEFGH = 30 = 2 3 2. Find the ratio of the perimeter of ABCD to the perimeter of EFGH.

Theorem 61: The ratio of the perimeters of two similar polygons equals the ratio of any pair of corresponding sides.

Given ∆BAT ~ ∆DOT OT = 15, BT = 12, TD = 9 Find the value of x(AO). AT = BT OT TD O 15 x + 15 = 12 15 9 D x = 5 B 12 9 T Hint: set up and use Means-Extremes Product Theorem.