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hw-9-9b-solns

# hw-9-9b-solns - Hamwaek C(K fa Lyn-om M 333 L FALL 20/0...

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Unformatted text preview: Hamwaek C(K fa Lyn-om M 333 L FALL 20 [/0 Exercises 4.6, #8 Solution: It is given that M, N, and P are the midpoints of sides KB, KC, and E of AABC, respectively. To Prove: The centroid of AMNP is the same point as the centroid of AABC. Proof: By the Midpoint Connection Theorem, Theorem 4.2.15, applied to AABC, MN M AC . By the Midpoint Connection Theorem, Theorem 4.2.15, applied to AABC, NP ll AB . Therefore, quadrilateral AMNP is a parallelogram. By Theorem 4.2.11, the diagonals of parallelogram AMNP bisect each other. Let R be the po_ir_1t_ where these diagonals intersect. Thus, R is the midpoint of MP . Therefore, NR is a median of AMNP . Since A - R - N , the median AN extending from vertex A of AABC anL the median ﬁ extending from vertex N of AMNP lie along the same line, AN . Recall that it was shown that MN H E . B By the Midpoint Connection Theorem, Theorem 4.2.15L applied to AABC, MP H BC . Therefore, quadrilateral MBNP is a parallelogram. By Theorem 4.2.11, the diagonals of parallelogram MBNP bisect each other. Let S be the point where these diagonals intersect. Thus, S is the midpoint of MN . Therefore, PS is a median of AMNP . A p C Since B - P- N , the median 5 extending from vertex B of AABC andH the median RS- extending from vertex P of AMNP lie along the same line, BP . Let T be the centroid of AABC. H H Then, T is the point of intersection of the two lines AN and BP which contain two of the medians of AABC . Since the two lines also contain two of the medians of AMNP , T is also the centroid of AMNP . Therefore, the centroid of AABC and the centroid of AMNP are both the same point T . QED §_E‘xa~c:3€s Li» 7 *9 Ba, Thaw (t.7-Z)1_t\l- {chivald '5 I 'p— § 3 64445an wlfzt ﬂu. ()P‘l’kocenfgh M 7Q i Giraumcenjfr‘ 4/ a fwlmogﬁ . ,. i S’tth‘I‘xLEqu/gvéane. "S TZurpmﬁcaloL lino) .. #17“ 09911795415 0n mgugnoﬂino a/ammas it ’Zucen‘faia‘ 1‘s ﬂqpouﬂ“ nfcmamence 4/& 7 Wm mdmm; 479V44)Ze , Sma ﬂu. Wm:_ i fisYLI/poi Woﬂmwﬂuwar’fﬂe WNW éwyzﬂwfalljrmﬂa 7444.722: zzq , , , 1mg ISM In 729‘ l‘ltéuﬁy‘é/Zd 757M EM“ IZqELJALueCannemvétf/nf'vé. , I’MJSIﬂL 72% 7‘me é—cdazuc ﬂu mﬁeldum Z“ 3 i i i % l i 4 ”Ll/d Farmewa 254144.34 71w. ,_ 7 . ofﬂwwwﬁ/LM ﬂu (7,!er €444ch 7L7Zrc szgh pAcL/‘M r5415» ﬂumﬁ'w/ Nosmzﬂp m/U/‘M WMV‘MIM 4 «gang 9.1 3 m units WAY TO GO: EUCLIDEANGEOMETRY OFTHE PLANE Exercises 4-.8 Ineach ofﬂtetdanglesinExcrcises l through4, Hismcorthoccntcran is the circumcenter. Locate the center and identify the length of the radius the nine-point circle associated with each triangle. Use a compass 00‘ draw the nine-point cimle for eachniangle. ' 1. A Ream; IS 0M; ...
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hw-9-9b-solns - Hamwaek C(K fa Lyn-om M 333 L FALL 20/0...

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