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Line ratios from similar polygons Complete each of the following statements: (a) If corresponding sides of two similar polygons are in the ratio of 4:3, then the ratio of their perime _ ters is __ . (b) The perimeters of two similar quadrilaterals are 30 and 24. If a side of the smaller quadrilateral _ is 8, the corresponding side of the larger is __ . _ (c) If each side of a pentagon is tripled and the angles remain the same, then each diagonal is __ .

n 1 0

4 . 3 p s s 30 . Then (b) Since the quadrilaterals are similar, and s 10. sr pr 8 24 (c) Tripled, since polygons are similar if their corresponding angles are congruent and their corresponding sides are in proportion. (a) Since the polygons are similar,

lirn HN( z ) =

The number b is called the y-coordinate of P (or the ordinate of P). Every point has a unique pair (a, b) of coordinates associated with it. EXAMPLES In Fig. 2-2, the coordinates of several points have been indicated. We have limited ourselves to integer coordinates only for simplicity.

-a-Iz- 1

In a problem, to prove that the product of the lengths of two segments equals the product of the lengths of another pair of segments, it is necessary to set up the proportion which will lead to the two equal products.

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which is not the inverse filter. Note that although &n) = h N ( n )* g ( n ) does not converge to S(n) as N + m, taking ) the limit o ~ B N ( zas N + co,we have

7.28 Proving an equal-products problem Prove that if two secants intersect outside a circle, the product of the lengths of one of the secants and its external segment equals the product of the lengths of the other secant and its external segment.

which is an all-pass filter, that is,

Given: Secants AB and AC. To Prove: AB AD AC AE Plan: Prove nABE , nACD to obtain AB AE . AC AD PROOF:

Fig. 2-2 Conversely, every pair (a, b) of real numbers is associated with a unique point in the plane. EXAMPLES In the coordinate system of Fig. 2-3, to nd the point having coordinates (3, 2), start at the origin O, move three units to the right and then two units upward. To nd the point with coordinates ( 2, 4), start at the origin O, move two units to the left and then four units upward. To nd the point with coordinates ( 1, 3), start from the origin, move one unit to the left and then three units downward. Given a coordinate system, the entire plane except for the points on the coordinate axes can be divided into four equal parts, called quadrants. All points with both coordinates positive form the rst quadrant, quadrant I, in the upper right-hand corner (see Fig. 2-4). Quadrant II consists of all points with negative x-coordinate and positive y-coordinate; quadrants III and IV are also shown in Fig. 2-4.

The first five samples of the unit sample response of a causal filter are h(0) = 3 h(1) = - 1 h(2) = 1

Reasons A segment may be drawn between any two points. Reflexive property. Angles inscribed in the same arc are congruent. Two triangles are similar if two angles of one triangle are congruent respectively to two angles of the other. 5. Corresponding sides of similar triangles are in proportion. AE 6. In a proportion, the product of the means equals the product of the extremes.

h(3) = 2

h(4) = 0

PRINCIPLE 1:

If it is known that the system function has two zeros and two poles, determine whether or not the filter is stable. The system function of this filter has the form

If two chords intersect within a circle, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other.

Fig. 2-3

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