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In Fig. 7-41, b
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PRINCIPLE 3:
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The length of the leg opposite the 60 angle equals the length of the leg opposite the 30 angle times the square root of 3.
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The length of the altitude of an equilateral triangle equals one-half the length of a side times the square root of 3. (Principle 4 is a corollary of Principle 2.)
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7.10B The 45 -45 -90 Triangle
A 45 -45 -90 triangle is one-half a square. In right triangle ABC (Fig. 7-42), c2 Hence, the ratio of the sides is a:a:c 1:1: 22. a2 a2 or c a 22
Fig. 7-42
Principles of the 45 -45 -90 Triangle
PRINCIPLE 5:
The length of a leg opposite a 45 angle equals one-half the length of the hypotenuse times the square root of 2.
1 2c
In Fig. 7-42, a
PRINCIPLE 6:
The length of the hypotenuse equals the length of a side times the square root of 2.
a 22.
In Fig. 7-42, c
CHAPTER 7 Similarity
Square Principle
PRINCIPLE 7:
In a square, the length of a diagonal equals the length of a side times the square root of 2.
s 22
In Fig. 7-42, d
SOLVED PROBLEMS
7.39 Applying principles 1 to 4 (a) If the length of the hypotenuse of a 30 -60 -90 triangle is 12, find the lengths of its legs [Fig.7-43(a)].
Fig. 7-43
(b) Each leg of an isosceles trapezoid has length 18. If the base angles are 60 and the upper base is 10, find the lengths of the altitude and the lower base [Fig. 7-43(b)].
Solutions
(a) By Principle 1, a (b) By Principle 2, h
1 2(12)
6. By Principle 2, b
1 2(12)
23 FD
6 23.
1 2(18)
1 2(18) 23
9 23. By Principle 1, AE
9; hence, b
7.40 Applying principles 5 and 6 (a) Find the length of the leg of an isosceles right triangle whose hypotenuse has length 28 [Fig. 7-44(a)].
Fig. 7-44
(b) An isosceles trapezoid has base angles measuring 45 . If the upper base has length 12 and the altitude has length 3, find the lengths of the lower base and each leg [Fig. 7-44(b)].
Solutions
(a) By Principle 5, a (b) By Principle 6, a
1 2 (28) 22
14 22. BE 3 and EF 12; hence, b 3 12 3 18.
3 22, AE
CHAPTER 7 Similarity
SUPPLEMENTARY PROBLEMS
7.1. Express each of the following ratios in lowest terms: (a) 20 cents to 5 cents (b) 5 dimes to 15 dimes (c) 30 lb to 25 lb (d) 20 to 14 (e) 27 min to 21 min (f) 50% to 25% (g) 15 to 75 (h) 33% to 77% (i) $2.20 to $3.30 (j) $.84 to $.96 (k) (l)
1 2 lb 1 22
(7.1) to
1 4lb
days to 31 days 2
1 yd to 12 yd
(m) 5 ft to 1ft 4 (n)
(o) 161 m to 51 m 2 2 (7.2) (i) 100 lb to 1 ton (j) $2 to 25 cents (k) 2 quarters to 3 dimes (l) 1 yd2 to 2 ft2 (7.3)
7.2. Express each of the following ratios in lowest terms: (a) 1 year to 2 months (b) 2 weeks to 5 days (c) 3 days to 3 weeks (d)
1 22
(e) 2 yd to 2 ft (f) (g)
1 23
yd to 2 ft to 9 in
1 12 ft
h to 20 min
(h) 2 g to 8 mg
7.3. Express each of the following ratios in lowest terms: (a) 20 cents to 30 cents to $1 (b) $3 to $1.50 to 25 cents (c) 1 quarter to 1 dime to 1 nickel (d) 1 day to 4 days to 1 week (e)
(f) 2 h to h to 15 min (g) 1 ton to 200 lb to 40 lb (h) 3 lb to 1 lb to 8 oz (i) 1 gal to 1 qt to 1 pt
day to 9 h to 3 h (7.4) (m) (n) (o) (p)
1 72 1 12 5 6 7 4
7.4. Express each of the following ratios in lowest terms: (a) 60 to 70 (b) 84 to 7 (c) 65 to 15 (d) 125 to 500 (e) 630 to 105 (f) 1760 to 990 (g) 0.7 to 2.1 (h) 0.36 to 0.24 (i) 0.002 to 0.007 (j) 0.055 to 0.005 (k) 6.4 to 8 (l) 144 to 2.4 to
1 22
to 10
to 12 3 to 8 (7.4)
7.5. Express each of the following ratios in lowest terms: (a) x to 3x (b) 15c to 5 (c) 11d to 22 (d) 2 r to D (e) ab to a2 (g) S3 to 6S2 (h) 9r2 to 6rt (i) x to 4x to 10x (j) 15y to 10y to 5y (k) x3 to x2 to x (l) 12w to 10w to 8w to 2w
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