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7.50. Find the lengths of the arms of a right triangle whose hypotenuse has length c if these arms have a ratio of (a) 3:4 and c 15; (b) 5:12 and c 26; (c) 8:15 and c 170; (d) 1: 2 and c 10. (7.34) 7.51. In a rectangle, find the length of the diagonal if its sides have lengths (a) 9 and 40; (b) 5 and 10. (7.33)

h(n)= [-2(;)"

+3(a)"]u(n)

7.52. In a rectangle, find the length of one side if the diagonal has length 15 and the other side has length (a) 9; (b) 5; (c) 10. (7.33) 7.53. Of triangles having sides with lengths as follows, which are right triangles (a) 33, 55, 44 (b) 120, 130, 50 (c) 4, 72, 82 (d) 25, 7, 24

f (x)Dx g(x) g(x)Dx f (x) + [g(x)]2 [g(x)]2 g(x)Dx f (x) f (x)Dx g(x) [g(x)]2

(b) To find the response of the system to x ( n ) = u ( n ) - u ( n - l o ) , we may proceed in one of two ways. First, we may perform the convolution of h ( n ) with x ( n ) :

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7.54. Is a triangle a right triangle if its sides have the ratio of (a) 3 : 4 : 5; (b) 2 : 3 : 4 7.55. Find the length of the altitude of an isosceles triangle if each of its two congruent sides has length 10 and its base has length (a) 12; (b) 16; (c) 18; (d) 10. (7.35)

Alternatively, noting that the input is a sum of two steps, we may find the step response of the system, s ( n ) , and then using linearity. write the response as

7.56. In a rhombus, find the length of a side if the diagonals have lengths (a) 18 and 24; (b) 4 and 8; (c) 6 and 6 23. (7.36) 7.57. In a rhombus, find the length of a diagonal if a side and the other diagonal have lengths, respectively, (a) 10 and 12; (b) 17 and 16; (c) 4 and 4; (d) 10 and 10 23. (7.36) 7.58. In isosceles trapezoid ABCD in Fig. 7-68 (a) Find a if b (b) Find h if b 32, b 24, b 20, and h 14, and a 8. 13. (c) Find b if a (d) Find b if a 15, b 6, b 10, and h 21, and h 12. 3 23. (7.37)

y ( n ) = s ( n ) - s ( n - 10) = [ 2 ( f ) " - ( : ) " ] u ( n ) - [ 2 ( f )

When k is positive, this is just Rule 4 ( 12). When k = 0, Dx (x k ) = Dx (x 0 ) = Dx (1) = 0 = 0 x 1 = kx k 1 Now assume k is negative; k = n, where n is positive. algebra By de nition, 1 x k = x n = n x By (1) of Problem 13.2, Dx (x k ) = Dx 1 xn = 1 xn

Fig. 7-68

11-10

( i ) n - l o ] u ( n- 10)

7.59. In a trapezoid ABCD in Fig. 7-69 (a) Find d if a (b) Find a if d 11, b 20, b 3, and c 12, and c 15. 36. (c) Find d if a (d) Find p if a 5, p 20, c 13, and c 28, and d 14. 17.

(c) With x(n) = (f)"u(n), note that .r(n) has the same form as one of the terms in the homogeneous solution. Therefore, the particular solution will not be of the form y,(n) = C(:)" as indicated in Table 1-2. If we were to substitute this particular solution into the difference equation, we would find that no value of C would work. As is the case when a root of the characteristic equation is of second order, the particular solution has the form

(7.37)

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