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Solved Problem 16.1 Henderson and Erin have decided to form a part nership. Henderson invests the following assets (shown at their agreed upon value) and he also transfers liabilities to the new rm.

PRINCIPLE 2:

In a right triangle, the length of either leg is the mean proportional between the length of the hypotenuse and the length of the projection of that leg on the hypotenuse.

Thus in right nABC (Fig. 7-31),

Fig. 7-32



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3 (a) By Principle 1, x (b) By Principle 1, x 8 y 12 x or x2 27, and x 3 23. By Principle 2, y , so y2 36 and y 9 3 y 20 8 , so y2 80 and y 4 25. and x 16. By Principle 2, y 4 4 6.

In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs.

Fig. 7-33

Henderson s Accounts Cash Accounts Receivable Merchandise Inventory Equipment Accounts Payable Notes Payable Value $17,500 7,000 10,000 4,200 3,500 3,600





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If c2 a2 b2 applies to the three sides of a triangle, then the triangle is a right triangle; but if c2 a2 b2, then the triangle is not a right triangle. In nABC, if c2 < a2 b2 where c is the longest side of the triangle, then the triangle is an acute triangle. Thus in Fig. 7-34, 92 < 62 82 (that is, 81 < 100); hence, nABC is an acute triangle. In nABC, if c2 > a2 b2 where c is the longest side of the triangle, then the triangle is an obtuse triangle. Thus in Fig. 7-35, 112 > 62 82 (that is, 121 > 100); hence, nABC is an obtuse triangle.

Fig. 7-34

Fig. 7-35

7.33 Finding the sides of a right triangle In Fig. 7-36, (a) find the length of hypotenuse c if a (c) find b if a 4 23 and c 8.

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Fig. 7-36

(a) c2 (b) a2 (c) b2 a2 c2 c2 b2 b2 a2 122 82 82 92 62 225 and c 28 and a 64 15. 2 27. 48 16 and b 4.

Erin agrees to invest $26,000 in cash. Record (a) Henderson s invest ment; (b) Erin s investment. Solution: (a) Cash Accounts Receivable Merchandise Inventory Equipment Accounts Payable Notes Payable Henderson, Capital (b) Cash Erin, Capital 17,500 7,000 10,000 4,200 3,500

(4 23)2

7.34 Ratios in a right triangle In a right triangle, the hypotenuse has length 20 and the ratio of the two arms is 3:4. Find each arm.

Let the lengths of the two arms be denoted by 3x and 4x. Then 202 (3x)2 (4x)2. Multiplying out, we get 400 9x2 16x2 or 400 25x2, and x 4; hence, the arms have lengths 12 and 16.

7.35 Applying the pythagorean theorem to an isosceles triangle Find the length of the altitude to the base of an isosceles triangle if the base is 8 and the equal sides are 12.

The altitude h of an isosceles triangle bisects the base (Fig. 7-37). Then h2 h 8 22. a2 (1b)2 2 122 42 128 and

7.36 Applying the pythagorean theorem to a rhombus In a rhombus, find (a) the length of a side s if the diagonals are 30 and 40; (b) the length of a diagonal d if a side is 26 and the other diagonal is 20.

The diagonals of a rhombus are perpendicular bisectors of each other; hence, 1 1 s2 (2d)2 (2d )2 in Fig. 7-38. (a) If d (b) If s 30 and d 26 and d 40, then s2 20, then 262 152 (2d)2

3,600

Fig. 7-37

Fig. 7-38

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