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(a) Its base has length 10 and the triangle is equal in area to a parallelogram whose base and altitude have lengths 15 and 8. (b) Its base has length 8 and the triangle is equal in area to a square whose diagonal has length 4. (c) Its base has length 12 and the triangle is equal in area to another triangle whose sides have lengths 6, 8, and 10. 9.17. In a triangle, find the length of (a) A side if the area is 40 and the altitude to that side has length 10 (b) An altitude if the area is 25 and the side to which the altitude is drawn has length 5 (c) A side if the area is 24 and the side has length 2 more than its altitude (d) A side if the area is 108 and the side and its altitude are in the ratio 3:2 (e) The altitude to a side of length 20, if the sides of the triangle have lengths 12, 16, and 20 (f) The altitude to a side of length 12 if another side and its altitude have lengths 10 and 15 (g) A side represented by 4x if the altitude to that side is represented by x (h) A side if the area is represented by x2 55, the side by 2x 7 and the area is 60 5 (9.4)

Use (2.1) to nd the squares of the sides, AB = ( 5 + 7)2 + ( 3 3)2 = 22 + ( 6)2 = 4 + 36 = 40 BC = ( 7 2)2 + (3 6)2 = 81 + 9 = 90 AC = ( 5 2)2 + ( 3 6)2 = 49 + 81 = 130 Since AB + BC = AC ,

The most common types of amplitude transformations are addition, multiplication, and scaling. Performing these operations is straightforward and involves only pointwise operations on the signal.

9.18. Find the area of an equilateral triangle if (a) a side has length 10; (b) the perimeter is 36; (c) an altitude has length 6; (d) an altitude has length 5 23; (e) a side has length 2b; (f) the perimeter is 12x; (g) an altitude has length 3r. (9.6)

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Scaling Amplitude scaling of a signal x ( n ) by a constant c is accomplished by multiplying every signal value by c:

The unit sample may be used to decompose an arbitrary signal x ( n ) into a sum of weighted and shifted unit samples as follows:

9.19. Find the area of a rhombus having an angle of 60 if (a) a side has length 2; (b) the shorter diagonal has length 7; (c) the longer diagonal has length 12; (d) the longer diagonal has length 6 23. (9.6) 9.20. Find the area of a regular hexagon if (a) a side is 4; (b) the radius of the circumscribed circle is 6; (c) the diameter of the circumscribed circle is 20. (9.6)

where each term in the sum, x(k)S(n - k ) , is a signal that has an amplitude of x ( k ) at time n = k and a value of zero for all other values of n . This decomposition is the discrete version of the svting property for continuous-time signals and is used in the derivation of the convolution sum.

2 2 2 2 2 2

9.21. Find the side of an equilateral triangle whose area equals (a) The sum of the areas of two equilateral triangles whose sides have lengths 9 and 12 (b) The difference of the areas of two equilateral triangles whose sides have lengths 17 and 15 (c) The area of a trapezoid whose bases have lengths 6 and 2 and whose altitude has length 9 23 (d) Twice the area of a right triangle having a hypotenuse of length 5 and an acute angle of measure 30

If h ( n ) is the unit sample response of an ideal low-pass filter with a cutoff frequency w, = n/4, find the frequency response of the filter that has a unit sample response g ( n ) = h(2n).

(9.6)

To find the frequency response of this system. we may work the problem in one of two ways. The first is to note that because the unit sample response of an ideal low-pass filter with a cutoff frequency w,. = n / 4 is

9.22. Find the area of trapezoid ABCD in Fig. 9-21, if: (a) b (b) m (c) AB (d) AB (e) AB 25, b 15, and h 6.9 30 , b 45 , b 24, and b 13, and b b 20 6 7 7

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