多边形的面积比是否为常数

多边形的面积比是否为常数 Öi$íÞ�ªu´Ñ�b? øø�Gç3íWä †\î ‡k: Ê �1Åbç`�ç}� (NCTM) ‡úò2J-çÞÑúïF%Ðíbç`‡qñ³, w2�ø_D�SóÉ, ñíÊ¿tçÞí:!?‰, Ý�x�hô� õðgM, /x��4í JAVA ˙�: �7.3 Understanding Ratio of Areas�� âk¥_½æÑ×¶Mí4�CçÞF�I, „{��25¬, 6ªœ.ñq„p, Ä7 �O JAVA ˙�U…‰Aø_xõð� hô� “¿� „p�Ö½gMíßæ‡...

Öi$íÞ�ªu´Ñ�b? øø�Gç3íWä †\î ‡k: Ê �1Åbç`�ç}� (NCTM) ‡úò2J-çÞÑúïF%Ðíbç`‡qñ³, w2�ø_D�SóÉ, ñíÊ¿tçÞí:!?‰, Ý�x�hô� õðgM, /x��4í JAVA ˙�: �7.3 Understanding Ratio of Areas�� âk¥_½æÑ×¶Mí4�CçÞF�I, „{��25¬, 6ªœ.ñq„p, Ä7 �O JAVA ˙�U…‰Aø_xõð� hô� “¿� „p�Ö½gMíßæ‡, õÑ�&Çê íªWj�5ø� 7.3. Understanding Ratios of Areas of Inscribed Figures Using Interactive Diagrams This example illustrates how students, using dynamic and interative geometric ﬁgures, can understand connections be- tween algebra and geometry, as described in the connections Standard. They can develop an understanding of how to jus- tify geometric relationships in a technological erivironment, as described in the Geometry Standard. …d: B: M u AB (¨,ª0Ñ 0.25 í}õ, õ, †² ¾ −→ OA+ −−→ OB + −→ OC + −→ OQ = −→ 0 , wŸJ O ÑŸõ, â}õt� −−→ OM = (1− r)−→OA+ r−−→OB, −−→ON = (1− r)−−→OB + r−→OC −→ OP = (1− r)−→OC + r−−→OD, −→OQ = (1− r)−−→OD + r−→OA Ä¤, −−→ OM + −−→ ON + −→ OP + −→ OQ = −→ OA+ −−→ OB + −→ OC + −−→ OD = −→ 0 ?¹sûi$ƒûÝõ²¾¸ÑÉíõó½¯�IA� B� C�Dûõíè™}�ÑA(x1, y1) B(x2, y2) C(x3, y3) D(x4, y4), †â}õt� −−→ OM = (1− r)(x1, y1) + r(x2, y2) = ((1− r)x1 + rx2, (1− r)y1 + ry2) −−→ ON = (1− r)(x2, y2) + r(x3, y3) = ((1− r)x2 + rx3, (1− r)y2 + ry3) −→ OP = (1− r)(x3, y3) + r(x4, y4) = ((1− r)x3 + rx4, (1− r)y3 + ry4) −→ OQ= (1− r)(x4, y4) + r(x1, y1) = ((1− r)x4 + rx1, (1− r)y4 + ry1) 70 bçfÈ 27»2‚ ¬92�6~ ûi$ MNPQ íÞ�= ∆MON Þ�+∆NOP Þ�+∆POQ Þ�+∆QOM Þ� = 1 2 ∣∣∣∣∣∣ (1− r)x1 + rx2 (1− r)y1 + ry2 (1− r)x2 + rx3 (1− r)y2 + ry3 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ (1− r)x2 + rx3 (1− r)y2 + ry3 (1− r)x3 + rx4 (1− r)y3 + ry4 ∣∣∣∣∣∣ + 1 2 ∣∣∣∣∣∣ (1−r)x3+rx4 (1−r)y3+ry4 (1−r)x4+rx1 (1−r)y4+ry1 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ (1−r)x4+rx1 (1−r)y4+ry1 (1−r)x1+rx2 (1−r)y1+ry2 ∣∣∣∣∣∣ (1) = 1 2 ∣∣∣∣∣∣ (1−r)x1 (1−r)y1 (1−r)x2 (1−r)y2 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ rx2 ry2 rx3 ry3 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ (1−r)x1 ry2 (1−r)x2 ry3 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ rx2 (1−r)y1 rx3 (1−r)y2 ∣∣∣∣∣∣ + 1 2 ∣∣∣∣∣∣ (1−r)x2 (1−r)y2 (1−r)x3 (1−r)y3 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ rx3 ry3 rx4 ry4 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ (1−r)x2 ry3 (1−r)x3 ry4 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ rx3 (1−r)y2 rx4 (1−r)y3 ∣∣∣∣∣∣ + 1 2 ∣∣∣∣∣∣ (1−r)x3 (1−r)y3 (1−r)x4 (1−r)y4 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ rx4 ry4 rx1 ry1 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ (1−r)x3 ry4 (1−r)x4 ry1 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ rx4 (1−r)y3 rx1 (1−r)y4 ∣∣∣∣∣∣ + 1 2 ∣∣∣∣∣∣ (1−r)x4 (1−r)y4 (1−r)x1 (1−r)y1 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ rx1 ry1 rx2 ry2 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ (1−r)x4 ry1 (1−r)x1 ry2 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ rx1 (1−r)y4 rx2 (1−r)y1 ∣∣∣∣∣∣ (2) ¾ (y“�, ”-íu: = (r2 + (1− r)2) { 1 2 ∣∣∣∣∣∣ x1 y1 x2 y2 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ x2 y2 x3 y3 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ x3 y3 x4 y4 ∣∣∣∣∣∣+ 1 2 ∣∣∣∣∣∣ x4 y4 x1 y1 ∣∣∣∣∣∣ } (1) �íû_W��·u¦Lv j²í²¾, wW��MîÑ£b, ]l�Þ�v, .à‹" úM� (2) �í©�í¬sá, uÿá�Ç(ªJ�ó¾ � Ä¤ûi$ MNPQ íÞ� = (r2 + (1− r)2){∆ AOB Þ� + ∆ BOC Þ� + ∆ COD Þ� + ∆ DOA Þ� } = (r2 + (1− r2)) ûi$ ABCD íÞ�, „H� Öi$íÞ�ªu´Ñ�b? – �Gç3íWä 71 ˙: Bbyhôüi$ Ç 8. L<üi$, ®i�ìª0u 0.25 í }õ, ©Aø_qüi$: Ç 9. L<üi$, ®i�ìª0u 0.25 í }õ, ©Aø_qüi$ ,Þs_Ç$íÞ�ªM.°� ½æ: L<íüi$ ABCDE, P� Q� R� S� T Ñ®i�ìª0u r í}õ, † ABCDE ¸ PQRST, qÕs_üi$íÞ�ªÑS? cq AP AB = BQ BC = CR CD = DS DE = ET EA = r Ç 10. 72 bçfÈ 27»2‚ ¬92�6~ J −→ OA = (x1, y1), −→ OC = (x3, y3), † |−→OA ∧−→OC| H[W�� ∣∣∣∣∣∣ x1 y1 x3 y3 ∣∣∣∣∣∣� üi$ PQRST Þ� = ∆ POQ Þ� + ∆ QOR Þ� + ROS Þ� + ∆ SOT Þ� + ∆ TOP Þ� = 1 2 {∣∣∣∣−→OP ∧−→OQ ∣∣∣∣+ ∣∣∣∣−→OQ ∧−→OR ∣∣∣∣+ ∣∣∣∣−→OR ∧ −→OS ∣∣∣∣+ ∣∣∣∣−→OS ∧ −→OT ∣∣∣∣+ ∣∣∣∣−→OT ∧ −→OP ∣∣∣∣ } â²¾}õt�, |−→OP ∧−→OQ|= |((1− r)−→OA+ r−−→OB) ∧ ((1− r))−−→OB + r−→OC| =(1−r)2|−→OA ∧−−→OB|+r2|−−→OB∧−→OC|+r(1−r)|−→OA∧−→OC| ·< |−−→OB ∧−−→OB| = 0 üi$ PQRST Þ� = 1 2 { (1− r)2|−→OA ∧−−→OB|+ r2|−−→OB ∧−→OC|+ r(1− r)|−→OA∧ −→OC| } +(1− r)2|−−→OB ∧−→OC|+ r2|−→OC ∧ −−→OD|+ r(1− r)|−−→OB ∧−−→OD| +(1− r)2|−→OC ∧ −−→OD|+ r2|−−→OD ∧−−→OE|+ r(1− r)|−→OC ∧ −−→OE| +(1− r)2|−−→OD ∧ −−→OE|+ r2|−−→OE ∧ −→OA|+ r(1− r)|−−→OD ∧ −→OA| +(1− r)2|−−→OE ∧−→OA|+ r2|−→OA ∧−−→OB|+ r(1− r)|−−→OE ∧ −−→OB| = (r2 + (1− r)2)(üi$ ABCDE íÞ�) +r(1−r)(|−→OA∧−→OC|+|−−→OB ∧ −−→OD|+|−→OC ∧ −−→OE|+|−−→OD ∧−→OA|+|−−→OE ∧ −−→OB|) Ä¤s_üi$íÞ�ªu.uìb� ¦²k r(1− r)(|−→OA∧ −→OC|+ |−−→OB ∧ −−→OD|+ |−→OC∧)−−→OE|+ |−−→OD ∧−→OA|+ |−−→OE ∧−−→OB| Düi$ ABCDE íÞ�ª, u´ÑìM� �: ÊL<ýi$³, ®i�ìª0u 0.25 í}õ, ©Aø_qýi$, s6Þ�ªE Í.�ì, ª"üi$íj¶T|¥W� O£ýi$®i�ìª0u 0.25 í}õ, F©Aø_ qýi$, JÉ��£ýi$íø_Ýõ, wÞ�ªEÍ�ì, ¡cÇ (11) (12), ,Þs_Ç íÞ�ª·u 0.812, D£ýi$íªMó°� ¥_„pD,Hè™�Sí„¶×_ó°, .Ç ;H� Öi$íÞ�ªu´Ñ�b? – �Gç3íWä 73 Ç 11. Ç 12. !x: çÞíç3, ñq ˘é“˙, c� ˘UW˙, "×¶Mí�Þ·AwÑvƒ7d†, ",Þ �ìª0õ�, Fd|VíqÕúi$íÞ�ªu�ìí, qÕûi$íÞ�ª6u�ìí�7/ Ê£üi$� £ýi$, à‹É��ø_Ýõ, qÕÖi$íÞ�ªEu�ìí� ku ˘Ü 7 g-˙, ÍL<üi$J,ÿ˜7� 7/„p´�óçí˚Ø , øOíçÞñqÜ 7 ˘J ;˝�˙ í²-, ku�S:!í?‰ÿ\Ïà7� ¥uø_óç|Hí�Gç3�SÖ‡, à ‹?éçÞAÐ «Ø� û˝, Fø}�'Öí-)� â¬í>áÅ�I}�¸çÍbÜ`>Í−ÿ`¤, Jbç`>í�Gç3hõ, �äB bªp NCTM í0ä, 6>áÅ�`¦×ç@àbçÍô×Ÿ`¤� ÏpO`¤ú…dí^ £� ‡DNû� ¡5d.: http://standards.nctm.org/document/eexamples/chap7/7.3/index.htm —…dT6L`kÅ�−œò�2ç—

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多边形的面积比是否为常数

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