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## Ratios, Proportions, and Similarity

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**Ratios, Proportions, and Similarity**Eleanor Roosevelt High School Chin-Sung Lin**ERHS Math Geometry**Ratio and Proportion Mr. Chin-Sung Lin**ERHS Math Geometry**A ratio is a comparison by division of two quantities that have the same units of measurement The ratio of two numbers, a and b, where b is not zero, is the number a/b e.g. AB = 4 cm and CD = 5 cm AB 4 cm 4 CD 5 cm 5 * A ratio has no units of measurement Definition of Ratio or 4 to 5 or 4 : 5 = = Mr. Chin-Sung Lin**ERHS Math Geometry**Since a ratio, like a fraction, is a comparison of two numbers by division, a ratio can be simplified by dividing each term of the ratio by a common factor e.g. AB = 20 cm and CD = 5 cm AB 20 cm 4 CD 5 cm 1 * A ratio is in simplest form (or lowest terms) when the terms of the ratio have no common factor greater than 1 Definition of Ratio or 4 to 1 or 4 : 1 = = Mr. Chin-Sung Lin**ERHS Math Geometry**A ratio can also be used to express the relationship among three or more numbers e.g. the measures of the angles of a triangle are 45, 60, and 75, the ratio of these measures can be written as 45 : 60 : 75 or, in lowest terms, ? Definition of Ratio Mr. Chin-Sung Lin**ERHS Math Geometry**A ratio can also be used to express the relationship among three or more numbers e.g. the measures of the angles of a triangle are 45, 60, and 75, the ratio of these measures can be written as 45 : 60 : 75 or, in lowest terms, 3 : 4 : 5 Definition of Ratio Mr. Chin-Sung Lin**ERHS Math Geometry**If the lengths of the sides of a triangle are in the ratio 3 : 3 : 4, and the perimeter of the triangle is 120 cm, Find the lengths of the sides Ratio Example Mr. Chin-Sung Lin**ERHS Math Geometry**If the lengths of the sides of a triangle are in the ratio 3 : 3 : 4, and the perimeter of the triangle is 120 cm, Find the lengths of the sides x: the greatest common factor three sides: 3x, 3x, and 4x 3x + 3x + 4x = 120 10x = 120, x = 12 The measures of the sides: 3(12), 3(12), and 4(12) or 36 cm, 36 cm, and 48 cm Ratio Example Mr. Chin-Sung Lin**ERHS Math Geometry**A rate is a comparison by division of two quantities that have different units of measurement e.g. A UFO moves 400 m in 2 seconds the rate of distance per second (speed) is distance 400 m time 2 s * A rate has unit of measurement Definition of Rate 200 m/s = = Mr. Chin-Sung Lin**ERHS Math Geometry**A proportion is a statement that two ratios are equal. It can be read as “a is to b as c is to d” a c b d * When three or more ratios are equal, we can write and extended proportion a c e g b d f h Definition of Proportion = or a : b = c : d = = = Mr. Chin-Sung Lin**ERHS Math Geometry**A proportion a : b = c : d a : b = c : d Definition of Extremes and Means extremes means Mr. Chin-Sung Lin**ERHS Math Geometry**When x is proportional to y and y = kx, k is called the constant of proportionality y x Definition of Constant of Proportionality is the constant of proportionality k = Mr. Chin-Sung Lin**ERHS Math Geometry**Properties ofProportions Mr. Chin-Sung Lin**ERHS Math Geometry**In a proportion, the product of the extremes is equal to the product of the means a c b d b ≠ 0, d ≠ 0 * The terms b and c of the proportion are called means, and the terms a and d are the extremes Cross-Product Property then a x d = b x c = Mr. Chin-Sung Lin**ERHS Math Geometry**In a proportion, the extremes may be interchanged a c d c b d b a b ≠ 0, d ≠ 0, a ≠ 0 Interchange Extremes Property then = = Mr. Chin-Sung Lin**ERHS Math Geometry**Interchange Means Property(Alternation Property) In a proportion, the means may be interchanged a c a b b d c d b ≠ 0, d ≠ 0, c ≠ 0 then = = Mr. Chin-Sung Lin**ERHS Math Geometry**The proportions are equivalent when inverse both ratios a c b d b d a c b ≠ 0, d ≠ 0, a ≠ 0, c ≠ 0 Inversion Property then = = Mr. Chin-Sung Lin**ERHS Math Geometry**If the products of two pairs of factors are equal, the factors of one pair can be the means and the factors of the other the extremes of a proportion c a b d b ≠ 0, d ≠ 0 a, b are the means, and c, d are the extremes Equal Factor Products Property a x b = c x d then = Mr. Chin-Sung Lin**ERHS Math Geometry**The proportions are equivalent when adding unity to both ratios a c a + b c + d b d b d b ≠ 0, d ≠ 0 Composition Property then = = Mr. Chin-Sung Lin**ERHS Math Geometry**The proportions are equivalent when subtracting unity to both ratios a c a - b c - d b d b d b ≠ 0, d ≠ 0 Division Property then = = Mr. Chin-Sung Lin**ERHS Math Geometry**Definition of Geometric Mean (Mean Proportional) Suppose a, x, and d are positive real numbers a x x d Then, x is called the geometric mean or mean proportional, between a and d then x2 = a d or x = √ad = Mr. Chin-Sung Lin**ERHS Math Geometry**Application Examples Mr. Chin-Sung Lin**ERHS Math Geometry**Maria has two job opportunities. If she works for a healthcare supplies store, she will be paid $60 daily by working 5 hours a day. If she works for a grocery store, she will be paid $320 weekly by working 8 hours a day and 5 days per week What is the pay rate for each job? What is the ratio between the pay rates of healthcare supplies store and grocery store? Example - Ratios and Rates Mr. Chin-Sung Lin**ERHS Math Geometry**Maria has two job opportunities. If she works for a healthcare supplies store, she will be paid $60 daily by working 5 hours a day. If she works for a grocery store, she will be paid $320 weekly by working 8 hours a day and 5 days per week What is the pay rate for each job? Healthcare: $12/hr, Grocery: $8/hr What is the ratio between the pay rates of healthcare supplies store and grocery store? Healthcare:Grocery = 12:8 = 3:2 Example - Ratios and Rates Mr. Chin-Sung Lin**ERHS Math Geometry**The perimeter of a rectangle is 48 cm. If the length and width of the rectangle are in the ratio of 2 to 1 What is the length of the rectangle? Example - Ratios Mr. Chin-Sung Lin**ERHS Math Geometry**The perimeter of a rectangle is 48 cm. If the length and width of the rectangle are in the ratio of 2 to 1 What is the length of the rectangle? Width: x Length: 2x 2 (2x + x) = 48 3x = 24, x = 8 2x = 16 Length is 16 cm Example - Ratios Mr. Chin-Sung Lin**ERHS Math Geometry**The measures of an exterior angle of a triangle and the adjacent interior angle are in the ratio 7 : 3. Find the measure of the exterior angle Example - Ratios Mr. Chin-Sung Lin**ERHS Math Geometry**The measures of an exterior angle of a triangle and the adjacent interior angle are in the ratio 7 : 3. Find the measure of the exterior angle Exterior angle: 7x Interior angle: 3x 7x + 3x = 180 10x = 180, x = 18 7x = 126 Measure of the exterior angle is 126 Example - Ratios Mr. Chin-Sung Lin**ERHS Math Geometry**Solve the proportions for x 2 x + 2 5 2x - 1 Example - Proportions = Mr. Chin-Sung Lin**ERHS Math Geometry**Solve the proportions for x 2 x + 2 5 2x - 1 2 ( 2x – 1) = 5 (x + 2) 4x – 2 = 5x + 10 -12 = x x = -12 Example - Proportions = Mr. Chin-Sung Lin**ERHS Math Geometry**If the geometric mean between x and 4x is 8, solve for x Example - Geometric Mean Mr. Chin-Sung Lin**ERHS Math Geometry**If the geometric mean between x and 4x is 8, solve for x x (4x) = 82 4x2 = 64 x2 = 16 x = 4 Example - Geometric Mean Mr. Chin-Sung Lin**ERHS Math Geometry**• 4x2 15y2 • 3 2 • 8x2 = 45 y2 • 4 3 • 15y2 2x2 • 5 4x • 2x 9y2 If Example - Properties of Proportions , which of the following = statements are true? Why? (x≠ 0, y ≠ 0) 4x + 9y 5y + 2x 9y 2x 9y 2x 4x - 9y 5y - 2x = = = = Mr. Chin-Sung Lin**ERHS Math Geometry**• 4x2 15y2 • 3 2 • 8x2 = 45 y2 • 4 3 • 15y2 2x2 • 5 4x • 2x 9y2 If Example - Properties of Proportions , which of the following = statements are true? Why? (x≠ 0, y ≠ 0) 4x + 9y 5y + 2x 9y 2x 9y 2x 4x - 9y 5y - 2x = = = = Mr. Chin-Sung Lin**ERHS Math Geometry**Midsegment Theorem Mr. Chin-Sung Lin**ERHS Math Geometry**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 Given: ∆ABC, D is the midpoint of AC, and E is the midpoint of BC Prove: DE || AB, and DE = ½ AB Midsegment Theorem C E D A B Mr. Chin-Sung Lin**ERHS Math Geometry**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 Given: ∆ABC, D is the midpoint of AC, and E is the midpoint of BC Prove: DE || AB, and DE = ½ AB Midsegment Theorem C E F D A B Mr. Chin-Sung Lin**ERHS Math Geometry**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 Given: ∆ABC, D is the midpoint of AC, and E is the midpoint of BC Prove: DE || AB, and DE = ½ AB Midsegment Theorem C E F D A B Mr. Chin-Sung Lin**ERHS Math Geometry**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 Given: ∆ABC, D is the midpoint of AC, and E is the midpoint of BC Prove: DE || AB, and DE = ½ AB Midsegment Theorem C E F D A B Mr. Chin-Sung Lin**ERHS Math Geometry**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 Given: ∆ABC, D is the midpoint of AC, and E is the midpoint of BC Prove: DE || AB, and DE = ½ AB Midsegment Theorem C E F D A B Mr. Chin-Sung Lin**ERHS Math Geometry**Divided Proportionally Theorem Mr. Chin-Sung Lin**ERHS Math Geometry**Two line segments are divided proportionally when the ratio of the lengths of the parts of one segment is equal to the ratio of the lengths of the parts of the other e.g., in ∆ABC, DC/AD = 2/1 EC/BE = 2/1 then, the points D and E divide AC and BC proportionally Definition of Divided Proportionally C E D A B Mr. Chin-Sung Lin**ERHS Math Geometry**If two line segments are divided proportionally, 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 Given: In ∆ABC, AD/DB = AE/EC Prove: AD/AB = AE/AC Divided Proportionally Theorem A E D B C Mr. Chin-Sung Lin**A**D E B C ERHS Math Geometry Statements Reasons 1. AD/DB = AE/EC1. Given 2. DB/AD = EC/AE 2. Inversion property 3. (DB+AD)/AD = (EC+AE)/AE 3. Composition property 4. DB+AD = AB, EC+AE = AC 4. Partition postulate 5. AB/AD = AC/AE 5. Substitution postulate 6. AD/AB = AE/AC 6. Inversion property Divided Proportionally Theorem Mr. Chin-Sung Lin**ERHS Math Geometry**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 proportionally Given: In ∆ABC, AD/AB = AE/AC Prove: AD/DB = AE/EC Converse of Divided Proportionally Theorem A E D B C Mr. Chin-Sung Lin**A**D E B C ERHS Math Geometry Statements Reasons 1. AD/AB = AE/AC 1. Given 2. AB/AD = AC/AE 2. Inversion property 3. (AB-AD)/AD = (AC-AE)/AE 3. Division property 4. AB-AD = DB, AC-AE = EC 4. Partition postulate 5. DB/AD = EC/AE 5. Substitution postulate 6. AD/DB = AE/EC 6. Inversion property Converse of Divided Proportionally Theorem Mr. Chin-Sung Lin**ERHS Math Geometry**Two line segments are divided proportionally if and only if 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 Divided Proportionally Theorem & Converse of Divided Proportionally Theorem A E D B C Mr. Chin-Sung Lin**ERHS Math Geometry**Application Examples Mr. Chin-Sung Lin**ERHS Math Geometry**In ∆ABC, D is the midpoint of AB and E is the midpoint of AC. BC = 7x+5, DE = 4x-2, BD = 2x+1, AC = 9x+1 Find DE, BC, BD, AB, AC, and AE Example - Divided Proportionally A E D B C Mr. Chin-Sung Lin**ERHS Math Geometry**In ∆ABC, D is the midpoint of AB and E is the midpoint of AC. BC = 7x+5, DE = 4x-2, BD = 2x+1, AC = 9x+1 Find DE, BC, BD, AB, AC, and AE 7x + 5 = 2 (4x – 2) 7x + 5 = 8x – 4 x = 9 DE = 4 (9) – 2 = 34 BC = 2 (34) = 68 BD = 2 (9) + 1 = 19 AB = 2 (19) = 38 AC = 9 (9) + 1 = 82 AE = 82/2 = 41 Example - Divided Proportionally A E D B C Mr. Chin-Sung Lin