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Chapter 12: Ratios , Proportion and Similarity Definitions: If the 2 means of a proportion are equal, the mean proportional is one of the means. (also called the geometric mean) Two line segments are divided proportionately when the ratio of the lengths of the parts of one segment is equal to the lengths of the parts of the other. Two Polygons are Similar if there is a one-to-one correspondence between vertices such that: All pairs of corresponding angles are congruent The ratios of the lengths of all pairs of corresponding sides are equal. If two polygons are similar, then their corresponding angles are congruent and their corresponding sides are in proportion. If two polygons have corresponding angles that are congruent and corresponding sides that are in proportion, then the polygons are similar. If two polygons have corresponding angles that are congruent and corresponding sides that are in proportion, then the polygons are similar. The ratio of any length in the transformed figure to the corresponding length in the original figure is called the scale factor (k) or constant of dilation. The ratio of the corresponding sides of similar figures is called the ratio of similitude. The point of concurrence of the medians of a triangle is called the Centroid Pythagorean Triples 3,4, 5 45-45-90 (1:1: 2 ) 30-60-90 (1: 3 : 2) Postulates, Theorems and Corollaries Theorem Corollary Corollary Corollary Theorem Theorem Theorem Theorem Theorem Theorem Corollary Theorem Theorem Theorem Theorem Theorem Theorem Theorem In a proportion, the product of the means is equal to the product of the extremes. In a proportion, the means may be interchanged. In a proportion, the extremes may be interchanged. If the products of 2 pairs of factors are equal, the factors of one pair can be the means, and the factors of the other can be the extremes of a proportion. A line segment joining the midpoints of two sides of a triangle is parallel to the third side and its length is one-half the length of the third side. If two line segments are divided proportionately, then the ratio of the length of a part of one segment to the length of the whole is equal to the ratio of the corresponding lengths of the other segment. If the ratio of the length of a part of one line segment to the length of the whole is equal to the ratio of the corresponding lengths of another line segment, then the two segments are divided proportionately. Two triangles are similar if the ratios of corresponding sides are equal. (SSS~) Two triangles are similar if the ratios of two corresponding sides are equal and the corresponding angles included between these sides are congruent. (SAS~) Two triangles are similar if three angles of one triangle are congruent to three corresponding angles of the other. Two triangles are similar if two angles of one triangle are congruent to three corresponding angles of the other. If two triangles are similar the lengths of corresponding altitudes have the same ratio of any two corresponding sides. If two triangles are similar the lengths of corresponding medians have the same ratio of any two corresponding sides. If two triangles are similar the lengths of corresponding angle bisectors have the same ratio of any two corresponding sides. Any two medians of a triangle intersect in a point that divides each median in the ratio of 2 : 1. The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to each other and to the original triangle. The altitude is the mean proportional between the measures of the segments of the hypotenuse. The length of each leg of a right triangle is the mean proportional between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg (projection of the leg). Pythagorean If a triangle is a right triangle, then the square of the length of the Theorem longest side is equal to the sum of the squares of the lengths of the other two sides (legs). Converse of If the square of one side a triangle is equal to the sum of the squares the of the lengths of the two other sides, then the triangle is a right Pythagorean triangle. Theorem