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Lesson 8.1 & 8.2

Lesson 8.1 & 8.2. Solving Problems with Ratio and Proportion. Today, we will learn to… …find and simplify ratios ...use proportions to solve problems. Ratio. A ratio is a comparison of two numbers written in simplest form. a a : b a to b b. Simplify the ratio.

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Lesson 8.1 & 8.2

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  1. Lesson 8.1 & 8.2 Solving Problems with Ratio and Proportion Today, we will learn to… …find and simplify ratios ...use proportions to solve problems

  2. Ratio A ratio is a comparison of two numbers written in simplest form. a a : b a to b b

  3. Simplify the ratio. K H D M D C M 1.2 m : 300 cm 2 : 3 200 cm : 300 cm 2. 2 km : 600 m 10 : 3 2000 m : 600 m 3. 10 mm : 5.5 cm 2 : 11 10 mm : 55 mm

  4. 4.In the diagram, DE : EF is 1 : 2 and DF = 45. Find DE and EF. D E F 1 2 x x DE = EF = 15 1x + 2x = 45 3x = 45 x = 15 30

  5. 5. In ΔABC, the measures of the angles are in the extended ratio of 3:4:5. Find the measures of the angles. What do we know about the angles of a triangle? 3x + 4x + 5x = 180 12x = 180 x = 15 °, °, ° 45 60 75

  6. 3x 2x 2x 3x 6. The perimeter of a rectangle is 70 cm. The ratio of the length to the width is 3 : 2. Find the length and the width of the rectangle. 3x+2x+3x+2x = 70 10x = 70 x = 7 Length is Width is 21 14

  7. 3x 2x 7.A triangle has an area of 48 m 2. The ratio of the base to the height is 2 : 3. Find the base and height. A = ½ bh 48 = ½ (2x)(3x) 48 = 3x2 16 = x2 8 m base is height is 4 = x 12 m

  8. Solve the proportion for x. 8. 2 8 7 x-2 2(x-2) = 56 2x - 4 = 56 2x = 60 x = 30

  9. 2in 180mi 9. On a map, 2 inch = 180 miles. Two cities are about 2 ¾ inches apart. Estimate the actual distance between them. 2 ¾in xmi 2x = 180(2¾) x = 247.5 miles

  10. 1/2 3 = 1/2 x = (3 )(30) 1/8 10. In a photograph taken from an airplane, a section of a city street is 3 1/2 inches long and 1/8 of an inch wide. If the actual street is 30 feet wide, how long is it? x 1/8 30 x = 840 feet

  11. = 11. AB : AC is 3 : 2. Find x. x+3 3 x+1 2 3(x+1) = 2(x+3) 3x+3 = 2x+6 x + 3 = 6 x = 3

  12. 12. Given MN MP find PQ. NO PQ = M = 4 14 P N 6 O Q 14-x 4 x 6 14-x ? 4x = 6(14-x) 4x = 84 - 6x x ? x = 8.4

  13. Given AB AD find DE. AC AE 13. = = 7 5 A 7+x 7 7 5(7+x) = 49 5 7+x ? 35+5x = 49 5x = 14 D B x = 2.8 x ? 2 C E

  14. x 420 mm 210 mm = x 14. Standard paper sizes are all over the world. The sizes all have the same width-to-length ratios. Two sizes of paper shown are A4 and A3. Find x. 210 x x 420 x2 = (210)(420) x ≈ 297 mm

  15. = 1 15. The batting average of a baseball player is the ratio of the number of hits to the number of official at-bats. In 1998, Sammy Sosa of the Chicago Cubs had 643 official at-bats and a batting average of .308. How many hits did Sammy Sosa get? x x = (643)(.308) .308 643 x = 198 hits

  16. = 16. A wheelchair ramp should have a slope of 1/12. If a ramp rises 2 feet, what is its run? ? 2 ft 2 ft 1 12 x What is its length? length2 = 22 + 242 x = (12)(2 ft) length2 = 4 + 576 x = 24 feet length2 = 580 length = 24.08 feet

  17. Geometric Mean The geometric mean of two positive numbers (a and b) is ….a x x b

  18. 35 x = x 175 Find the geometric mean of the given numbers. 35 and 175 x2= 35(175) x ≈ 78.3

  19. Lesson 8.3Similar Polygons Today, we will learn to… …identify similar polygons ...use similar polygons

  20. AB BC AC Two polygons are similar ifall corresponding angles are congruent and corresponding sides are proportional. ΔABC ~Δ XYZ if A B C  X Y Z and XZ YZ XY

  21. C G F B E H A D CDGH ADEH ABEF BCFG ABCD ~ EFGH A E, B F, C G, D H Statement of Proportionality

  22. Scale FactorThe scale factor is the ratio of the lengths of two corresponding sides.

  23. 6 8 10 2 3 1. Are the triangles similar? If they are, find the scale factor and write a statement of similarity. 15 9 12 Yes, the scale factor is XAR ~  __ __ __ M N T

  24. 4.5 6 9 3 4 2. Are the triangles similar? If they are, find the scale factor and write a statement of similarity. 8 12 6 Yes, the scale factor is LMN ~  __ __ __ T P O

  25. A = D x 12 12 y 15 10 3 2 15 10 15 10 y 12 10 15 C F 12 x E B = = 12 15 x 4. ΔABC ~ ΔDEF Scale Factor? x = 18 y 10 y = 8 12

  26. B B x x 8 8 9 y 18 E A A C C 12 12 9 x y 12 18 8 18 8 18 9 F D y = = The triangles are similar. Find x and y. 5. 12 x C 12 A 8 x 8 B Map the triangles to find corresponding sides. y = 27 x = 4

  27. 5 3 x = 6. RSTU ~ LMNO. Find the following. 125 mT = mS = 55 x = 4 x 2.4

  28. = 7. You have a 3.5 inch by 5 inch photo that you want to enlarge. You want the enlargement to be 16 inches wide. How long will it be? 3.5 16 5 x 3.5x = (16)(5) x = 22.9 ≈ 23 inches

  29. 3 4 3 4 36 48 12 in. 16 in. A triangular work of art and the frame around it are similar equilateral triangles. 8. Find the ratio of the artwork to the frame. 9. Find the ratio of the perimeters. (artwork : frame)

  30. 7 4 4 5 4 5 22 27.5 220 275 5 8.75 10. Find the ratio of corresponding sides. The rectangles are similar. 11. Find the ratio of the perimeters.

  31. Theorem 8.1If 2 polygons are similar, then the ratio of the perimeters is __________ the ratio of corresponding side lengths. equal to

  32. = patio pool 12. The patio around a pool is similar to the pool. The perimeter of the pool is 96 feet. The ratio of the patio to the pool is 3 to 2. Find the perimeter of the patio. 3 x 2x = (3)(96) 2 96 x = 144 feet

  33. Turn to page 145 in your workbook!

  34. Lesson 8.4 Proving Triangles are Similar Triangles Today, we will learn to… …identify similar triangles ...use similar triangles

  35. Postulate 25 Angle-Angle (AA) Similarity Two triangles are similar if 2 pairs of corresponding angles are congruent.

  36. Determine whether the triangles are similar. If they are, write a similarity statement. 1. M R 27˚ 80˚ 80˚ L 65˚ 35˚ L N 35˚ 65˚ T S M L N ΔRTS ~ Δ____

  37. Determine whether the triangles are similar. If they are, write a similarity statement. 2. G 27˚ H L 27˚ K J G K J ΔGLH ~ Δ____

  38. 4. If the triangles are similar, write a similarity statement. 47˚ 31˚ not similar

  39. 5. If the triangles are similar, write a similarity statement. 43˚ not similar

  40. 5 3 3 7 = 6. The triangles are similar, find x. 3 5 7 x y 2 2 2 x y 3y = 14 3x = 10 x ≈ 3.33 y ≈ 4.67

  41. 15 9 A B 9 15 C 25 18 x E D 8. The triangles are similar. Find x. = x 25 x = 15

  42. Z W X Are the triangles similar? If they are, write a similarity statement. T Y X XTY XZW ~  Not ~

  43. Are the triangles similar? If they are, write a similarity statement. 40 75 Not ~ ABD ~  BCE

  44. Lesson 8.5Proving Triangles areSimilar Triangles Today, we will learn to… …use similarity theorems to prove that two triangles are similar

  45. Theorem 8.2 Side-Side-Side (SSS) SimilarityIf all three corresponding sides are proportional, then the triangles are similar.

  46. 12 15 18 A D 12 8 10 15 C F 12 18 E B Determine whether the triangles are similar. If they are, write a similarity statement. 1. 8 12 10 scale factor? 3:2 ΔACB ~ Δ____ by _____ DFE SSS

  47. Theorem 8.3 Side-Angle-Side (SAS) Similarity If two sides are proportional and the angles between them are congruent, then the triangles are similar.

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