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18; hence, the angles measure 54 and 36 . 110 or x 22; hence, the angles measure 66 and 44 .
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180, so 5x
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7.6 Three angles having a fixed ratio Three angles are in the ratio of 4 : 3 : 2. Find the angles if (a) the first and the third are supplementary; (b) the angles are the three angles of a triangle.
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Let the measures of the angles be 4x, 3x, and 2x. Then: (a) 4x (b) 4x 2x 3x 180, so that 6x 2x 180, so 9x 180 for x 180 or x 30; hence, the angles measure 120 , 90 , and 60 . 20; hence, the angles measure 80 , 60 , and 40 .
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4 A proportion is an equality of two ratios. Thus, 2: 5 4: 10 (or 2 10) is a proportion. 5 The fourth term of a proportion is the fourth proportional to the other three taken in order. Thus in 2 : 3 4 : x, x is the fourth proportional to 2, 3, and 4. The means of a proportion are its middle terms, that is, its second and third terms. The extremes of a proportion are its outside terms, that is, its first and fourth terms. Thus in a : b c : d, the means are b and c, and the extremes are a and d.
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If the two means of a proportion are the same, either mean is the mean proportional between the first and fourth terms. Thus in 9:3 3 :1, 3 is the mean proportional between 9 and 1.
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7.2A Proportion Principles
PRINCIPLE 1:
In any proportion, the product of the means equals the product of the extremes.
c : d, then ad bc.
Thus if a: b
PRINCIPLE 2:
If the product of two numbers equals the product of two other numbers, either pair may be made the means of a proportion and the other pair may be made the extremes.
5y, then x : y 5 : 3 or y : x 3 : 5 or 3 : y 5 : x or 5: x 3 : y.
Thus if 3x
7.2B Methods of Changing a Proportion into an Equivalent Proportion
PRINCIPLE 3:
(Inversion method) A proportion may be changed into an equivalent proportion by inverting each ratio.
x 4 , then 5 1 5 . 4
1 Thus if x
PRINCIPLE 4:
(Alternation method) A proportion may be changed into an equivalent proportion by interchanging the means or by interchanging the extremes.
y x , then y 2 3 2 or 2 3 y x.
Thus if
PRINCIPLE 5:
(Addition method) A proportion may be changed into an equivalent proportion by adding terms in each ratio to obtain new first and third terms.
c a b , then b d c d d . If x 2 2 9 x , then 1 2 10 . 1
Thus if
PRINCIPLE 6:
(Subtraction method) A proportion may be changed into an equivalent proportion by subtracting terms in each ratio to obtain new first and third terms.
a b c , then b d c d d . If x 3 3 9 x , then 1 3 8 . 1
Thus if
7.2C Other Proportion Principles
PRINCIPLE 7:
If any three terms of one proportion equal the corresponding three terms of another proportion, the remaining terms are equal.
3 x and 5 4 3 , then y 5 4.
x Thus if y
PRINCIPLE 8:
In a series of equal ratios, the sum of any of the numerators is to the sum of the corresponding denominators as any numerator is to its denominator.
c d e a , then f b c d e f a x y . If b 4 y 5 3 x 3 , then 1 y 4 y 5 3 1 3 x 3 or 1 10 3 . 1
Thus if
SOLVED PROBLEMS
7.7 Finding unknowns in proportions Solve the following proportions for x: (a) x : 4 (b) 3: x 6: 8 x : 27 (c) x :5 3 (d) x 2 5 2x :(x 3) (e) x 3 2x 3 5 7 x 2 (f) 4 x 2
CHAPTER 7 Similarity
Solutions
(a) Since 4(6) (b) Since x2 8x, 8x 3(27), x2 x(x 3(5), 2x 3) (x 24 or x 81 or x 3. 9. x2 3x. Then x2 7x 0, so x 0 or 7.
(c) Since 5(2x) (d) Since 2x (e) Since 3(2x (f) Since 4(7)
3), we have 10x 15 or x
1 72.
5x, we have 6x 2)(x
5x, so x x2
9. 4. Then x2 32, so x 4 22.
2), we have 28
7.8 Finding fourth proportionals to three given numbers Find the fourth proportional to (a) 2, 4, 6; (b) 4, 2, 6; (c) 1, 3, 4; (d) b, d, c. 2
Solutions
(a) We have 2:4 (b) We have 4: 2 (c) We have 2:3 (d) We have b:d
6 : x, so 2x 6 : x, so 4x 4: x, so 2x c: x, so bx
24 or x 12 or x 12 or x cd or x
12. 3. 24. cd/b.
7.9 Finding the mean proportional to two given numbers Find the positive mean proportional x between (a) 5 and 20; (b) 1 and 8. 2 9
Solutions
(a) We have 5:x (b) We have 2:x
x: 20, so x2 x: 9 , so x2
8 4 9
100 or x or x
2 3.
7.10 Changing equal products into proportions (a) Form a proportion whose fourth term is x and such that 2bx (b) Find the ratio x to y if ay
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