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2.14 Find the value of k such that (3, k) is equidistant from (1, 2) and (6, 7). 2.15 (a) Are the three points A(1, 0), B( 7 , 4) and C(7, 8) collinear (that is, all on the same line) [Hint: If A, B, C form a triangle, 2 the sum of two sides, AB+BC must be greater than the third side, AC. If B lies between A and C on a line, AB+BC = AC.] (b) Are the three points A( 5, 7), B(0, 1), and C(10, 11) collinear 2.16 Find the midpoints of the line segments with the following endpoints: and (5, 3). (a) (1, 1) and (7, 5); (b) ( 3 , 4) and (1,0); (c) ( 2, 1) 2

where a and b are real, and la I < 1. Find the relationship between a and h that must exist if the frequency response is to have a constant magnitude for all w , that is, I H ( ~ ~ =) 1 * I

x(n) = (;)"u(n)

9.31. Provide the proofs requested in Fig. 9-24.

1 + h2 2b cos w ( b + e-jo)(h + eJ") I H ( ~ ' "=I ~ ) I + a2 - 2a cos w ( 1 - ae-JU)(l- aeju)

(9.9)

Therefore, it follows that I H(e'")12 = 1 if and only if b = -a. if x(n) = ( i ) " u ( n )Y ( e j U is given by , ) With a = and h =

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2.17 Find the point (a, b) such that (3, 5) is the midpoint of the line segment connecting (a, b) and (1, 2). 2.18 Prove by use of coordinates that the line segment joining the midpoints of two sides of a triangle is one-half the length of the third side. 2.19 Prove by use of coordinates that the line segments joining the midpoints of opposite sides of a quadrilateral ABCD bisect each other.

Fig. 9-24

= What we observe from this example is that although I H (eJU)l 1, the nonlinear phase has a significant effect on the values of the input sequence.

9.32. Prove each of the following: (a) A median divides a triangle into two triangles having equal areas.

Show that the group delay of a linear shift-invariant system with a frequency response H(ejw)may be expressed as H R ( ~ ~ , ) GR ( d w ) H I (ejW)G(ejw)

(9.10)

t h ( ~= )

17.1 Verify Rolle s theorem for f (x) = x 3 3x 2 x + 3 on the interval [1, 3].

where HR(eJw) Hl(ejw)are the real and imaginary parts of H(ejw),respectively, and G R ( e j Wand and ) G r ( e j w ) the real and imaginary parts of the DTFT of nh(n). are

(b) Triangles are equal in area if they have a common base and their vertices lie in a line parallel to the base. (c) In a triangle, if lines are drawn from a vertex to the trisection points of the opposite sides, the area of the triangle is trisected. (d) In trapezoid ABCD, base AD is twice base BC. If M is the midpoint of AD, then ABCM and BCDM are parallelograms which are equal in area. 9.33. (a) In ^ABC, E is a point on BM, the median to AC. Prove that area(^BEA) area(^BEC).

+j~;(ej")

(b) In ^ABC, Q is a point on BC, M is the midpoint of AB, and P is the midpoint of AC. Prove that area(^BQM) area(^PQC) area(quadrilateral APQM). area (^ACD). (c) In quadrilateral ABCD, diagonal AC bisects diagonal BD. Prove that area(^ABC)

where HA(eJW) the derivative of the real part of H(eiw) and H;(ejW)is the derivative of the imaginary part, the is group delay may be written as

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