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Given: EG i BD, EF i BC To Prove: FG i CD Plan: Prove that FG divides AC and AD proportionately. PROOF:

2. A line (segment) parallel to one side of a triangle divides the other two sides proportionately. 3. Substitution postulate 4. If a line divides two sides of a triangle proportionately, it is parallel to the third side.



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($28,000 $4,000) / 32,000 hours = 0.75 times 7,200 hours = $5,400 Solved Problem 15.4 Equipment costing $9,600, with an estimated scrap value of $1,600, was bought on July, 1, 1999. The equipment is to be depreciated by the straight-line method for a period of 10 years. The company s scal year is January through December. Show the journal en try to record the equipment s cost on the balance sheet for December 1999. Solution: Equipment Less: Accumulated Depreciation $9,600 400





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Similar polygons are polygons whose corresponding angles are congruent and whose corresponding sides are in proportion. Similar polygons have the same shape although not necessarily the same size. The symbol for similar is ,. The notation nABC , nA B C is read triangle ABC is similar to triangle A-prime B-prime C-prime. As in the case of congruent triangles, corresponding sides of similar triangles are opposite congruent angles. (Note that corresponding sides and angles are usually designated by the same letter and primes.) In Fig. 7-8 nABC , nA B C because m/A and m/A 37 a ar m/B b br m/B c cr or 53 6 3 m/C 8 4 m/C 10 5 90

Fig. 7-8

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Schemas are employed with XML to identify a set of components for use in XML documents and to provide the rules for their correct combination. A schema is an XML-based alternative to a DTD that describes the structure of an XML document. Like DTDs, schemas de ne the following: Elements that appear in a document Attributes that appear in a document Elements that are child elements The sequence in which the child elements can appear The number of child elements Whether an element is empty or can include text Default values for attributes

Fig. 7-9

See Fig. 18-14.

Fig. 18-14

18.14 Combining transformations Name the single transformation that does the same thing as the combination of: (a) P(x, y) A Pr(x 7, y 2) and then Qr(x, y) A Qs(x 3, y x, y) 5, y 1) 3, y 2) 5) (b) P(x, y) A Pr( x, y) and then Qr(x, y) A Qs(x (c) P(x, y) A Pr( x, y) and then Qr(x, y) A Qs(4 (d) P(x, y) A Pr(y, x) and then Qr(x, y) A Qs(x

(a) R(x, y) A Rs((x (b) R(x, y) A Rs( x (c) R(x, y) A Rs(4 (d) R(x, y) A Rs(y ( 5, 7) 3, y x), x 3, (y 2) y) 1) Rs(4 x, y) 2) 5) Rs(x 10, y 3)

$9,200

18.15 Recognizing rigid motions Name the transformation that takes ^ABC to (a) ^ArBrCr, (b) ^AsBsCs , and (c) ^A-B-C- as illustrated in Fig. 18-15.

Fig. 18-15

(a) The triangle has been reflected across the y axis and then moved up 1 space, so P(x, y)APr( x, y (b) The triangle has been reflected across the line y P(x, y) A Pr(x 2, 7 y). 1).

(c) The triangle has been rotated clockwise around the origin 90 and then moved up 8 spaces and to the right 6 spaces, so P(x, y) A Pr(y 6, x 8).

18.16 Naming rigid motions Name the transformation that (a) Rotates everything around the origin 180 , then moves everything up 3 spaces (b) Reflects across x 4, then slides everything down 2 spaces

(c) Rotates everything 90 clockwise around the origin, then reflects across the y axis (d) Rotates around the origin 90 counter-clockwise, then slides to the left 5 spaces

($9,600 $1,600) divided by 10 years = $800 depreciation per year 1/2 year ( July 1 to Dec.31) times $800 per year = $400

(a) P(x, y) A Pr( x, (b) P(x, y) A Pr(8 (d) P(x, y) A Pr( y y x, y 3) 2)

(c) P(x, y) A Pr( y, x) 5, x)

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