1 / 41

Geometric Similarities

Geometric Similarities. Math 416. Geometric Similarities Time Frame. 1) Similarity Correspondence 2) Proportionality (SSS) (Side-side-side) 3) Proportionality (SAS) (side-angle-side) 4) Similarity Postulates 5) Deductions 6) Dimensions 7) Three Dimensions. Similarity Correspondence.

benson
Download Presentation

Geometric Similarities

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. Geometric Similarities Math 416

  2. Geometric Similarities Time Frame • 1) Similarity Correspondence • 2) Proportionality (SSS) (Side-side-side) • 3) Proportionality (SAS) (side-angle-side) • 4) Similarity Postulates • 5) Deductions • 6) Dimensions • 7) Three Dimensions

  3. Similarity Correspondence • Similarity – Two shapes are said to be similar if they have the same angles and their sides are proportional • Note – we see shape by angles & we see size with side length

  4. Consider A Similar & Why? D 8 100 95 24 16 X 5 W 100 95 10 15 85 80 80 85 B 32 C Z Y 20

  5. Proportionality (SSS) • We say the two shapes are similar because their angles are the same and their sides are proportional • We can note corresponding points • A  X • D  W • B  Y • C  Z

  6. Angles • We note corresponding angles • < ADC = < XWZ (95°) • < DCB = < WZY (85°) • < CBA = < ZYX (80°) • < BAD = < YXW (100°)

  7. Notes • Hence we would say ADCB XWZY • Hence we note corresponding angles • < ADC = < XWZ • < DCB = < WZY • < CBA = < ZYX • < BAD = < YXW

  8. Proportionality • Next is proportionality which we will state as a fraction • AD=8DC=16CB=32BA=24 XW 5 WZ 10 ZY 20 YX 15 • What is the proportion (not in a fraction)? • 8/5 which is reduced to 1.6

  9. Question #1 • Identify the similar figures and state the similarity relationship, side proportion and angle equality T A BIG MED MED B Z BIG C SMALL SMALL C

  10. Notes for Solution • By observing you need to establish the relationship. • Look at angles or side lengths • Important: An important trick when comparing angles and sides is that the biggest angles is always across the biggest side, the smallest from the smallest and medium from the medium.

  11. Solution #1 • Triangle ABC ˜ TCZ • AB = BC = CA TC CZ ZT • < ABC = < TCZ < BCA = <CZT < CAB = <ZTC

  12. Important Note • Make sure the middle angle letters are all different because the middle letter is the actual angle that you are looking at. • AC = CA • < ACB = < BCA • Both the above are the same

  13. Question #2 R K SMALL MED MED T MED L MED SMALL X Q With isosceles (or equilateral triangles) you may get two (or three) different answers). However, you are only required to provide one.

  14. Solution a for #2 • The question is to identify similar figures and state the similarity relationship, side proportion and graph equality. • QK = KT = QT RX XL RL • < QKT = < RXL < KTQ = < XLR < TQK = < LRX QKT ˜ RXL

  15. Solution b for #2 • You can also have another solution • Triangle QKT is still congruent to RLX • QK = KT = QT RL LX RX • < QKT = < RLX < KTQ = < LXR < TQK = XRL

  16. More Notes  • There are other ways of establishing similarity in triangles • At this point we will abandon reality for simple effective but not accurate drawings of triangles… (it is not to scale). • Please complete #1 a – o • For Question #3, again, state similarity relationship, side proportion and angle equality.

  17. Question #3 T A 35 45 21 18 Q B 30 R 27 C If the three sides are proportional to the corresponding three sides in the other triangle, the two will be similar.

  18. Solution Notes • You need to check… • SMALL with SMALL • MEDIUM with MEDIUM • BIG with BIG

  19. Solution #3 • ABC ˜ QRT Small Small Big Med Med Big • 18 = 21 = 27 • 35 45 • 0.6 = 0.6 = 0.6; YES SIMILAR

  20. Proportionality SAS • We can also show similarity in triangles if we can find two set corresponding sides proportional and the contained angles equal; we can determine similarity X A 15 18 14° 42 14° Y 35 Z B C

  21. Question #4 • Show if the triangle is similar • Solution… since • <BAC = <XYZ = 14° • 18 = 42 15 35 = 6/5 6/5 • BAC ˜ XYZ • Notice BAC = Small, Angle, Big & compared to Small, Angle, Big

  22. Triangle Similarity Postulates • There are three main postulates we use to state similarity • SSS all corresponding sides proportional • SAS two sets of corresponding sides and the contained angle are equal • AA two angles (the third is automatically equal since in a triangle, the interior angle must add up to 180°) are equal

  23. Example #1 • Why are the following statements true? • QPT ˜ ZXA AA X Q 84° 42° 54° 84° 54° P T A Z

  24. Example #2 • Why are the following statements true? • KTR ˜ PMN SAS M R 51° 18 16 27 51° T K N 24 P Solution: since 24/16 = 27/18

  25. Example #3 K A 16 12 6 24 B C T 9 32 P Since 16 = 24 = 32 6 9 12 8/3 = 8/3 = 8/3 S S S

  26. Parallel Lines • Facts: If two tranversals intersect three parallel lines, the segments between the lines are proportional a c b d Therefore, a = c b d

  27. Notes • Also note that… C • BC = 1 AC 2 B • A •

  28. Parallel and the Triangle • If a parallel line to a side of a triangle intersects the other two sides it creates two similar triangles A Therefore, ABE ˜ ACD B E C D

  29. Question #1 3 9 • 3 = 9 • x • 3x = 36 • x = 12 4 x Find x

  30. Question #2 A We note BE // CD Thus, ABE ˜ ACD AB = BE = AE AC CD AD x = 50 = AE x+40 150 AD x E B 50 40 C D 150

  31. Question #2 Sol’n Con’t • We only need x = 50 x+40 150 • 150x = 50(x+40) • 150x = 50x + 2000 • 100x = 2000 • x = 20

  32. Proportion Ratio • Consider 1 10 1 10

  33. Dimensions

  34. Dimensions • In general in 1D if a:b then in 2D a2 : b2 • Ex. In 1D if 5:3 then in 2D? • In 2D then 25:9 • You can go backwards by using square root • Ex. In 2D if 36:49 then in 1D • 6:7

  35. 3D or Volume • Consider 1 5 1 5 1 5

  36. 3D or Volume

  37. 3D or Volume • In general in 1D if a:b then in 3D… • Then in 3D a3 : b3 • Ex. In 1D if 6:7 then in 3D • In 3d 216:343 • You can go backwards by using the cube root • Ex. In 3D if 27:8 then in 1D • In 1D 3:2

  38. Practice Complete the following 1 D Length 2 D Area 3 D Volume 2:9 4:81 8:729 2:11 4:121 8:1331 25:9 125:27 5:3

  39. 3D Question #1 • Two spheres have a volume ratio of 64:125. If the radius of the large one is 11cm, what is the radius of the small one? • Big Small • r 11 x 3D Ratio 64:125 1D Ratio 4:5 Thus 4 = x 5 11 5x = 44 x = 8.8

  40. 3D Question #2 V = ? V=200m3 A base = 16 m2 A Base = 100m2

  41. Question #2 Solution Big Small Area of Base 100 16 Volume 200 x 1 Ratio 10 / 4 2 Ratio 100/16 3 Ratio 1000/64 Thus 200 = 1000 x 64 1000x = 12800 x = 12.8

More Related