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AB is a tangent.
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Fig. 6-67
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Prove each of the following: (a) The base angles of an inscribed trapezoid are congruent. (b) A parallelogram inscribed in a circle is a rectangle. (c) In a circle, parallel chords intercept equal arcs. (d) Diagonals drawn from a vertex of a regular inscribed pentagon trisect the vertex angle. (e) If a tangent through a vertex of an inscribed triangle is parallel to its opposite side, the triangle is isosceles.
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(6.18)
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7.1 Ratios
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Ratios are used to compare quantities by division. The ratio of two quantities is the first divided by the second. A ratio is an abstract number, that is, a number without a unit of measure. Thus, the ratio of 10 ft to 5 ft is 10 ft 5 ft, which equals 2. A ratio can be expressed in the following ways: (1) using a colon, as in 3:4; (2) using to as in 3 to 4; (3) as a common fraction, as in 3; (4) as a decimal, 0.75; and (5) as a percent, 75%. 4 The quantities involved in a ratio must have the same unit. A ratio should be simplified by reducing to lowest terms and eliminating fractions. Thus to find the ratio of 1 ft to 4 in, we first change the foot to 12 in, and then take the ratio of 12 in to 4 in; the result is a ratio of 3 to 1, or 3. Also, the ratio of 21 : 1 would be restated 2 2 as 5 :1 or 5. The ratio of three or more quantities may be expressed as a continued ratio. Thus, the ratio of $2 to $3 to $5 is the continued ratio 2:3:5. This enlarged ratio is a combination of three separate ratios; these are 2:3, 3:5, and 2:5. Throughout this chapter, readers should use a calculator whenever they choose.
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7.1 Ratio of two quantities with the same unit Express each of the following ratios in lowest terms: (a) 15 to 3 ; (b) $1.25 to $5; (c) 21 years to 2 2 years.
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Solutions
15 5 3 1.25 1 (b) 5 4 21 2 5 (c) 2 4 (a)
7.2 Ratio of two quantities with different units Express each of the following ratios in lowest terms: (a) 2 years to 3 months; (b) 80 cents to $3.20.
Solutions
(a) 2 years to 3 months (b) 80 cents to $3.20 24 months to 3 months 80 cents to 320 cents 80 320 24 3 1 4 8
CHAPTER 7 Similarity
7.3 Continued ratio of three quantities Express each of the following ratios in lowest terms: (a) 1 gal to 2 qt to 2 pt; (b) 1 ton to 1lb to 8 oz.
Solutions
(a) 1 gal to 2 qt to 2 pt (b) 1 ton to 1 lb to 8 oz 4 qt to 2 qt to 1 qt
4:2:1 2000 :1 : 2
2000 lb to 1 lb to 2 lb
4000 : 2 : 1
7.4 Numerical and algebraic ratios Express each of the following ratios in lowest terms: (a) 50 to 60; (b) 6.3 to 0.9; (c) 12 to 3; (d) 2x to 8 5x; (e) 5s2 to s3; (f) x to 5x to 7x.
Solutions
50 60 6.3 (b) 0.9 (a) (c) 12 5 6 7 3 8 32 (d) 2x 5x 5s2 (e) 3 s 2 5 5 s 1:5: 7
(f) x : 5x : 7x
7.5 Using ratios in angle problems If two angles are in the ratio of 3 : 2, find the angles if (a) they are adjacent and form an angle measuring 40 ; (b) they are acute angles of a right triangle; (c) they are two angles of a triangle whose third angle measures 70 .
Solutions
Let the measures of the angles be 3x and 2x. Then: (a) 3x (b) 3x (c) 3x 2x 2x 2x 40, so that 5x 90, so 5x 70 40 or x 8; hence, the angles measure 24 and 16 .
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