1 / 42

Chapter 8 Similarity

Chapter 8 Similarity. 232 Geometry BEHS Mrs. Prescott. Ratios. Ratio – a comparison of 2 numbers in the same unit of measure Example: 2 females to 3 males 2 to 3 2:3. Simplifying Ratios. Reduce using common factors as you would simplify any fraction Pages 461-462 12. 16. 22. 26.

maylin
Télécharger la présentation

Chapter 8 Similarity

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 8 Similarity 232 Geometry BEHS Mrs. Prescott

  2. Ratios • Ratio – a comparison of 2 numbers in the same unit of measure • Example: 2 females to 3 males • 2 to 3 • 2:3

  3. Simplifying Ratios • Reduce using common factors as you would simplify any fraction • Pages 461-462 12. 16. 22. 26.

  4. Proportions • Proportion – a statement equating 2 ratios • Example: • Also written - • 3:6 = 5:10 means extremes

  5. Proportions • The product of the means is equal to the product of the extremes • 3:6 = 5:10 • The cross products in a proportion are always equal which is why we cross multiply to solve a proportion 65 310 • 310 = 65 • 30 = 30

  6. Solve the Proportion -4x -4x

  7. Properties of Proportions • Cross Product Property • The Product of the extremes equals the product of the means. • Reciprocal Property • If two ratios are equal, then their reciprocals are also equal.

  8. Page 463 #56. -7k -7k 3 3

  9. Extended Ratios • Simplify the extended ratio of the 4 sides of the quadrilateral. • 20:16:40:36 = • 5:4:10:9 20cm 16cm 36cm 40cm

  10. Examples: Side lengths are 9 ft, 12 ft, 9 ft, and 12 ft. • The perimeter of a parallelogram is 42ft, and ratio of 2 of its unequal sides is 3:4. Find each side length. 4 sides – 3:4: 3:4 or 3x:4x:3x:4x 3x + 4x + 3x + 4x = 42 14x = 42 x = 3 • If the extended ratio of the angles of a triangle are 5:6:7, find each angle measure. 5x:6x:7x 5x + 6x + 7x = 180 18x = 180 x = 10 Angle measures are 50º, 60º, 70º,

  11. Section 8.2 Problem Solving in Geometry with Proportions Page 465 232 Geometry BEHS, Mrs. Prescott

  12. Properties of Proportions from Section 8.1

  13. Additional Properties of Proportions: Ex. ¾ =9/12, then 3/9=4/12 • If • If Ex. ¾ = 9/12, then (3+4)/4 = (9+12)/12 7/4 = 21/12

  14. Examples: True True or False. Complete the statement. False 15 m+9

  15. Example – page 469 J K 2 5 26. P Q 7 x 9 X + 5 S -7x -7x

  16. Geometry Mean - definition The geometric mean of 2 positive numbers a and b is the positive number x such that

  17. Geometry Mean-Example Find the geometric mean of 3 and 48. √ √

  18. Geometry Mean-Example Find the geometric mean between 6 and 15.

  19. Section 8.3 Similar Polygons Madeleine Wood Page 473 232 Geometry BEHS, Mrs. Prescott

  20. Similar Polygons • Definition: Similar Polygons – 2 polygons such that: • their corresponding angles are congruent • the lengths of their corresponding sides are proportional • Symbol:

  21. Similar Polygons Example: • You can see that the corresponding angles are congruent. • Corresponding sides are in proportion means that the ratios of every 2 pairs of corresponding sides are equal. They all reduce to ½ , which is called the scale factor.

  22. Similarity Statement: ABCD~EFGH The order of the vertices indicates which angles correspond and which segments correspond in the similarity statement. To write a proportionality statement, write the ratios of all pairs of corresponding sides. List all pairs of congruent corresponding angles. Write the proportionality statement.

  23. Examples: Page 475 Rotate to make it easier to match the corresponding sides 8 3 15 10 No, because 15/10≠ 8/3 Yes, all corr. ∡s are ≅ and the ratios of all corr. sides are =.

  24. Examples: Page 475. Given: TUVW~ABCD ∡A≅∡T, ∡B≅∡U, ∡C≅∡V, ∡D≅∡W, and 9 B A Scale factor of ABCD to TUVW 6 70º D D C Scale factor of TUVW to ABCD is 5/3 15 U T V W 23

  25. Examples: Page 475 6. 7. Find all missing segment lengths and angle measures. 9 B A 110º 110º 6 6 70º 70º D C y 15 U T 110º 110º =10 10 x 70º 70º V W 23

  26. Given that ΔRST~ΔJKL. find the value of x and y. 70º 70º 50º (2w-5)º 70 + 50 + (2w – 5) = 180 115 + 2w = 180 2w = 65 w = 32.5

  27. Complete each. • Find the scale factor of the triangles. • Find the lengths of the missing segment lengths. • Find the perimeter of each triangle. • Find the ratio of the perimeter of the 2 triangles.

  28. Similar Polygon Perimeter Theorem If 2 polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.

  29. Section 8.4

  30. Given that ΔRSV~ΔRTU, answer each of the following. • Write the statement of proportionality. • m∡RSV=_______, m∡U= _______ • RS = ______ RT=______ SV= ______ RV=______ R 80º 5 80˚ 70˚ S V 8 10 15 5 70º T U 4 4 12

  31. AA ~ Postulate • If 2 angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. • Given that ∡D ≅ ∡A and ∡C ≅ ∡F, then ΔABC ~ ΔDEF. D A C B F E

  32. Chapter 8- Section 8.5 Proving Triangles are Similar Page 488 232 Geometry Mrs. Prescott BEHS

  33. SSS ~ Theorem • If the lengths of the corresponding sides of two triangles are proportional, then the two triangles are similar. D A 9 6 6 4 C B F E 8 12

  34. SAS ~ Theorem If an angle of one triangle is congruent to an angle of a 2nd triangle, and the lengths of the sides including these angle are proportional, then the two triangles are similar. D A 6 9 38º C 8 B 38º F 12 E

  35. Practice Problems: page 492 #2-5 • AA~ Post., ΔABC ~ΔDEF • SAS~ Thm., ΔABC ~ΔDFE • Both ΔJKL and ΔMNP; SSS~ • Ratio = 1:6, the triangles are similar by SSS~

  36. Chapter 8- Section 8.6 Proportions and Similar Triangles Page 498 232 Geometry Mrs. Prescott BEHS

  37. Triangle Proportionality Theorem A line that intersects 2 sides of a triangle is parallel to the 3rd side if and only if it divides the 2 sides proportionally. R 5 4 S V 8 10 and T U

  38. Examples: 3

  39. Theorem: If 3 parallel lines intersect two transversals, then they divide the transversals proportionally. A B D C F E

  40. Example: Solve for x and y.

  41. Theorem: • If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other 2 sides. D 3 9 G 4 F E 12

  42. Find the value of A. Example:

More Related