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Key Skills

Find the image of the segment with endpoints (–2, 6) and (4, 2) after the dilation. (–1, 3). . . (2, 1). 1. 1. x ,. y. D = ( x , y ) =. 2. 2. 8.1 Dilations and Scale Factors. Key Skills. Draw a dilation on a coordinate plane.

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Key Skills

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  1. Find the image of the segment with endpoints (–2, 6) and (4, 2) after the dilation (–1, 3)   (2, 1) . 1 1 x, y D = (x, y) = 2 2 8.1 Dilations and Scale Factors Key Skills Draw a dilation on a coordinate plane. The endpoints of the image are (–1, 3) and (2, 1).

  2. Dilate the triangle about A, using the scale factor 3. A  preimage TOC 8.1 Dilations and Scale Factors Key Skills Draw a dilation in a plane. image Each point on the image is 3 times as far from the center of dilation as the corresponding point on the preimage.

  3. 8.2 Similar Polygons Objectives • Similar Polygons

  4. 8.2 Similar Polygons Theorems, Postulates, & Definitions Similar Figures 8.2.1: Two figures are similar if and only if one is congruent to the image of the other by a dilation. Polygon Similarity Postulate 8.2.2: Two polygons are similar if and only if there is correspondence between their sides and angles so that: 1. Each pair of corresponding angles is congruent; 2. Each pair of corresponding sides is proportional.

  5. Cross-Multiplication Property 8.2.3: c c a b a and b and d  0, then ad = bc. If = = d d b a b Reciprocal Property 8.2.4: d = . and a, b, c, and d  0, then If c 8.2 Similar Polygons Theorems, Postulates, & Definitions

  6. Exchange Property 8.2.5: c a a a b c = . and a, b, c, and d  0, then = If = d b b c d d "Add-One" Property 8.2.6: a + b c + d and b and d  0, then . If = b d 8.2 Similar Polygons Theorems, Postulates, & Definitions

  7. 30 18 3 , so sides are also proportional. = = 40 24 4 8.2 Similar Polygons Key Skills Determine whether polygons are similar. Are these rectangles similar? Explain. Yes; all angles are 90, so corresponding angles are congruent.

  8. 15 20 so 15x = (18)(20) = 360. = x 18 TOC 8.2 Similar Polygons Key Skills Use proportions to find side lengths of similar figures. These triangles are similar. Find x. Therefore, x = 24 feet.

  9. 8.3 Triangle Similarity Theorems, Postulates, & Definitions AA (Angle-Angle) Similarity Postulate 8.3.1: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. SSS (Side-Side-Side) Similarity Theorem 8.3.2: If the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar.

  10. 8.3 Triangle Similarity Theorems, Postulates, & Definitions SAS (Side-Angle-Side) Similarity Theorem 8.3.3: If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.

  11. 9 12 3 = = 6 8 2 Since , the two pairs of sides that include those angles are proportional. TOC 8.3 Triangle Similarity Key Skills Use the AA Similarity Postulate and the SSS and SAS Similarity Theorems to determine triangle similarity. Are these two triangles similar? 1 2 by Vertical Angles Theorem. So the triangles are similar by the SAS Similarity Theorem.

  12. 8.3 Triangle Similarity Theorems, Postulates, & Definitions AA (Angle-Angle) Similarity Postulate 8.3.1: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. SSS (Side-Side-Side) Similarity Theorem 8.3.2: If the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar.

  13. 8.3 Triangle Similarity Theorems, Postulates, & Definitions SAS (Side-Angle-Side) Similarity Theorem 8.3.3: If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.

  14. 9 12 3 = = 6 8 2 Since , the two pairs of sides that include those angles are proportional. TOC 8.3 Triangle Similarity Key Skills Use the AA Similarity Postulate and the SSS and SAS Similarity Theorems to determine triangle similarity. Are these two triangles similar? 1 2 by Vertical Angles Theorem. So the triangles are similar by the SAS Similarity Theorem.

  15. 8.5 Indirect Measurement and Additional Similarity Theorems Theorems, Postulates, & Definitions Proportional Altitudes Theorem 8.5.1: If two triangles are similar, then their corresponding altitudes have the same ratio as their corresponding sides. Proportional Medians Theorem 8.5.2: If two triangles are similar, then their corresponding medians have the same ratio as their corresponding sides.

  16. 8.5 Indirect Measurement and Additional Similarity Theorems Theorems, Postulates, & Definitions Proportional Angle Bisectors Theorem 8.5.3: If two triangles are similar, then their corresponding angle bisectors have the same ratio as their corresponding sides. Proportional Segments Theorem 8.5.4: An angle bisector of a triangle divides the opposite side into two segments that have the same ratio as the other two sides.

  17. 105  40x = 7875  x = 196.875 = x 40 75 8.5 Indirect Measurement and Additional Similarity Theorems Key Skills Use similar triangles to measure distance indirectly. Estimate the width of the lake. The triangles each contain a right angle and 1  2. Triangles are similar (AA Similarity Postulate) The lake is nearly 200 meters wide.

  18. 12 8  12x = 72  x = 6 feet = 9 x TOC 8.5 Indirect Measurement and Additional Similarity Theorems Key Skills Use similarity theorems to solve problems involving altitudes and medians of triangles. These triangles are similar. Find x. Since the triangles are similar, corresponding medians have the same ratio as the corresponding sides of the triangles.

  19. The ratio of the sides of two similar triangles is 3 . 5 9 32 . The ratio of their areas is = 52 25 8.6 Area and Volume Ratios Key Skills Find the ratios of the areas of similar figures. Find the ratio of the areas of the two triangles.

  20. . The ratio of the radii of two similar cones is 3 7 27 . The ratio of their volumes is = 33 343 73 TOC 8.6 Area and Volume Ratios Key Skills Find the ratios of the volumes of similar solids. Find the ratio of the volumes of the two cones.

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