Проектирование уроков развивающего обучения по геометрии в старших классах: Методические указания

:JKLJ:L?HJBBNMGDPBCB=?HF?LJBB IJH?DLBJH...

Author: Донцов В.Н. | Скляднев С.А.

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2

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4

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Mjhd fZl_fZlbdb – ijbdeZ^gZy fbdjh^b^ZdlbdZ p_ehklgZy mihjy^hq_ggZy \ ijhkljZgkl\_ b \j_f_gb ]_g_lbq_kdZy de_lhqdZ \ fZdjhkljmdlmj_ijhp_kkZh[mq_gbydhlhjh_ij_ke_^m_lljb_^bgmxp_ev` =benhj^ Klj_fe_gb_ d jZpbhgZevghfm b wdhghfghfm j_r_gbx \k_]^Z oZjZdl_jbah\Zeh ^_yl_evghklv djmiguo fZl_fZlbdh\ ijhreh]h b gZklhys_]h GZijbf_j ba\_klguc hl_q_kl\_gguc fZl_fZlbd < =emrdh\ hlf_qZe ³P_ev fZl_fZlbdb – wlh \k_]^Z ihemq_gb_ g_ dZdh]h-gb[m^v Z bf_ggh kZfh]h baysgh]h kZfh]h ijhklh]h j_r_gby” (1965). >bnn_j_gpbZevgh-ikboheh]bq_kdb_ bkke_^h\Zgby ijhn_kkhjZ <: Djml_pdh]h ijb\_eb _]h d \u\h^m ³?keb ^ey mq_gbdh\ kh kj_^gbfb kihkh[ghklyfb p_ev aZdexqZ_lky \ lhf qlh[u j_rblv aZ^Zqm lh ^ey kihkh[guo d fZl_fZlbd_ hgZ aZdexqZ_lky \ lhf qlh[u j_rblv _z gZbemqrbfgZb[he__wdhghfgufkihkh[hf1.” <:Djml_pdbcIkboheh]byfZl_fZlbq_kdbokihkh[ghkl_crdhevgbdh\– F–K 1

5

I_^Z]h]bq_kdhfm ^hklb`_gbx k\_joaZ^Zqb kbkl_fu mjhdh\ ih knhjfmebjh\Zgghc l_f_ fh]ml [ulv ih^qbg_gu ke_^mxsb_ qZklgu_ ^b^Zdlbq_kdb_p_eb 1) ^hdZaZlvbkbkl_fZlbabjh\Zlv\nhjf_mdjmigzgguo^b^Zdlbq_kdbo [ehdh\ k\hckl\Z ijZ\bevghc n–m]hevghc ibjZfb^u ^ey jZaebqguo dhgdj_lguoagZq_gbcngZijbf_jn =4; 5; 6; 2) l_hj_lbq_kdb h[h[sblv wlb k\hckl\Z ^ey kemqZy ijhba\hevgh]h n (n∈N, n ≥ 3 bknhjfmebjh\Zlvfg_fhgbq_kdbcZe]hjblfboaZihfbgZgby 3) ijhdhgljhebjh\Zlvbhp_gblvmjh\gbh\eZ^_gbymqZsbfbky Z f_lh^hfijyfhm]hevgh]hl_ljZw^jZ [ f_lh^hf\hkoh^ys_]hZgZebaZ \ Ze]_[jZbq_kdbff_lh^hf\kl_j_hf_ljbb ] f_lh^Zfb ihkljh_gby ijh_dpbhgguo bah[jZ`_gbc ijZ\bevghc 2 ibjZfb^u ; 4) jZkrbjblv m]em[blv b mihjy^hqblv ]_hf_ljbq_kdbc l_aZmjmk mqZsboky q_j_a h[_ki_q_gb_ mk\h_gby bfb gh\uo hihjguo nhjfme oZjZdl_jbamxsbo h[h[szggu_ ljb]hghf_ljbq_kdb_ k\hckl\Z fgh`_kl\Z ijZ\bevguoibjZfb^ 5) jZkdjulv h[jZapu ijbeh`_gbc nhjfbjm_fh]h f_lh^Z ijb j_r_gbb klZg^Zjlguobg_klZg^ZjlguoaZ^Zq Ijhn_kkbhgZevgZy dhfi_l_glghklv mqbl_ey fZl_fZlbdb dZd i_^Z]h]bq_kdh]hf_g_^`_jZijhy\ey_lky\h[_ki_q_gbb_^bgkl\Zh[mq_gbyb \hkiblZgby mqZsboky hgph\Ijh_dlbjh\Zgb_kbkl_fumjhdh\hihjguo aZ^Zq\dmjk_]_hf_ljbb–deZkkh\–
6

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H;?ibr_\Z<BDjmibqMqblvrdhevgbdh\mqblvkyfZl_fZlbd_ Nhjfbjh\Zgb_ijbzfh\mq_[^_yl_evghklbDg^eymqbl_ey–F–K-16; <G>hgph\Mq_[gZykZfhhj]ZgbaZpbyklm^_glh\–fZl_fZlbdh\–<> dZd nZdlhjmki_\Z_fhklb:\lhj_ndZg^^bkk–E977.–k 3

7

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8

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1.2.

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10

5. W\jbklbq_kdbc ihbkd mqZsbfbky b nhjfmebjh\dZ mqbl_e_f Ze]hjblfZ eh]bq_kdh]haZihfbgZgby::Kfbjgh\<YEym^bk \k_o^_kylbnhjfmek ihfhsvxfg_fhgbq_kdh]hdjm]Zjbk 6. Nhjfmebjh\dZ mqbl_e_f gh\h]h \b^Z hjb_glbjh\hqghc hkgh\u IY =Zevi_jbg GN LZeuabgZ fZl_fZlbq_kdhc ^_yl_evghklb aZdexqzgghc \ ihgylbbijZ\bevghcibjZfb^u 7. Q_luj_h[h[szgguolbiZhkgh\guodhg\_j]_glguoaZ^Zq>`=benhj^ h \uqbke_gbb m]eh\uo we_f_glh\ ijZ\bevghc n–m]hevghc ibjZfb^u ih ke_^mxs_cmdjmig_gghcko_f_ n/n I II III IV

>Zgh GZclb α θ Β Φ

α

θ

Β

Φ

– 7 4 5

7 – 3 9

4 3 – 6

5 9 6 –

ey ijZ\bevghc n–m]hevghc ibjZfb^u kZfhklhyl_evgh ih h[jZapZf^Zggufmqbl_e_f ^hdZaZlvk\hckl\ZlZ[ebpu? 2) J_a_j\gh_aZ^Zgb_j_rblvaZ^ZqmZ[ ba 9,wlZi IhklZgh\dZ\ZjbZlb\gh]h^hfZrg_]haZ^Zgbyf_lh^jZkihjy`_gby qzldhcbgkljmdpbb ?]hkh^_j`Zgb_fh`_l[ulv\ujZ`_ghke_^mxsbfb wibkl_fbq_kdbfblj_[h\ZgbyfbjZa\b\Zxs_]hoZjZdl_jZXD;Z[Zgkdbc 1) Ih dhgki_dlm \umqblv k\hckl\Z ljb]hghf_ljbq_kdb_ nhjfmeu ijZ\bevghcn–m]hevghcibjZfb^uZe]hjblfboeh]bq_kdh]haZihfbgZgbyb ^hdZaZl_evkl\Z 2) >hdZaZlvk\hckl\ZbalZ[ebpu? . 3) Ih\lhjblvljb]hghf_ljbq_kdb_k\hckl\Zijyfhm]hevgh]hl_ljZw^jZ k p_evxih^]hlh\dbdfZl_fZlbq_kdhfm^bdlZglm 4) J_rblv • \k_fihaZ^Zqgbdmih^j_^Zdpb_cFBKdZgZ\b[20@ • `_eZxsbf kf Z[ l_dkl aZjZg__ \u\_rb\Z_lky \ fZl_fZlbq_kdhfm]hed_deZkkZ

11

• ijbah\mxaZ^Zqmkf_ ^eyn=5. 5) K^_eZlv jZ[hlm gZ^ hrb[dZfb \ ij_^u^ms_c ^hfZrg_c jZ[hl_ b \ deZkkghckZfhklhyl_evghcjZ[hl_ 9,,wlZi AZdexq_gb_ mjhdZ f_lh^ djZldh]h h[h[s_gby dhgljhey b dhjj_dpbb _jdZqB:AbfgyyG<DmavfbgZ:DFZjdh\Z::J_Zg 1) l_hj_lbq_kdb_blh]bmjhdZ-e_dpbb 2) hp_gdZ mjh\g_c ih^]hlh\e_gghklb mqZsboky mk\h_gby bfb l_fu bo j_ne_dkbb khljm^gbq_kl\Z b bgl_ee_dlmZevghc khklyaZl_evghklb dhffmgbdZlb\ghc b j_q_\hc dmevlmju B: Abfgyy :: E_hglv_\ :: Klheyj mqZsbokygZmjhd_ 3) h[ty\e_gb_Zj]mf_glbjh\Zgguohlf_lhdihmjhqgh]h[ZeeZ 4) k[hj l_ljZ^_c k ^hfZrg_c jZ[hlhc gZ kiehrgmx beb \u[hjhqgmx ijh\_jdm Ijb\_^zggucklbevm]em[e_ggh]hp_e_ijh_dlbjh\ZgbymjhdZfZl_fZlbdb\ dmjk_³F_lh^bdb´fh`_l[ulvgZeh`_ggZijh_dlu^jm]bolbih\mjhdh\

12

F?LH>IJYFHM=HEVGH=HL?LJ:W>J: L_ljZw^j \k_ ]jZgb dhlhjh]h – ijyfhm]hevgu_ lj_m]hevgbdb gZau\Zxlijyfhm]hevguf Imklv\ijyfhm]hevghfl_ljZw^j_AXBC: AC⊥ ie;%& ; XB⊥ ie$%& ;

∠CXB=α; ∠AXC=β; ∠AXB=γ; A –\_ebqbgZ^\m]jZggh]hm]eZijbj_[j_XA, B –\_ebqbgZ^\m]jZggh]hm]eZijbj_[j_XBjbk F_lh^ijyfhm]hevgh]h l_ljZw^jZ–wlhdhf[bgbjh\Zggucf_lh^j_r_gbykl_j_hf_ljbq_kdboaZ^Zq \dexqZxsbc\k_[y

dhgkljmdlb\gh-]_hf_ljbq_kdh_

\uqe_g_gb_

\

kljmdlmj_

ijhkljZgkl\_gghcnb]mjuijyfhm]hevgh]hl_ljZw^jZ Ze]_[jZbq_kdmx nhjfmebjh\dm gZ hkgh\_ fg_fhgbq_kdh]h djm]Z jbk h^ghc beb g_kdhevdbo ljb]hghf_ljbq_kdbo nhjfme \ujZ`Zxsbo k\yav f_`^m aZ^Zggufb b bkdhfufb m]eh\ufb we_f_glZfb lZ[e : Fg_fhgbq_kdh_

ijZ\beh

khklhbl

ba

ke_^mxs_c

Ze]hjblfbq_kdhc

ihke_^h\Zl_evghklbrZ]h\ 1°JZkiheh`bfihke_^h\Zl_evghklv\_ebqbgm]eh\A, γ, B, α, 90°, β^ey lj_o]jZggh]hm]eZk\_jrbghcX\iylbk_dlhjZofg_fhgbq_kdh]hdjm]Z Z ijhimklb\ijbwlhfijyfhc^\m]jZggucm]hekj_[jhfXCb [ aZf_gb\ \_ebqbgu _]h iehkdbo m]eh\ α, β g_ ijhlb\he_`Zsbo ijyfhfm ^\m]jZgghfmm]emkj_[jhfXC gZ(90°-α), (90°-β)khhl\_lkl\_ggh :γ<°-α, 90°-β; 2° \uqbkeb\ dhkbgmk dZ`^h]h we_f_glZ djm]Z dZd ijhba\_^_gb_ dhlZg]_gkh\ ^\mo khk_^gbo k gbf m]eh\uo we_f_glh\ ihemqbf i_j\u_ iylv hihjguonhjfme 3°\gh\v\uqbkeb\dhkbgmkdZ`^h]hwe_f_glZdjm]ZdZdijhba\_^_gb_ kbgmkh\ ^\mo g_ khk_^gbo k gbf m]eh\uo we_f_glh\ ihemqbf gh\u_ iylv hihjguonhjfme5. >hdZaZl_evkl\Zkf\f_lh^ihkh[bbKdey^g_\K:>hgph\<G Ijh_dlbjh\Zgb_kbkl_fumjhdh\hihjguoaZ^Zq\dmjk_]_hf_ljbb– 11 deZkkh\–
13

14

LZ[ebpZ: LJB=HGHF?LJBQ?KDB?KJ: i i

1.

Ij_^\Zjbl_evgh

HdhgqZl_evgh

cos A = ctg γ ⋅ ctg ( 90 ° − β ) cosΑ = ctgγ ⋅ tgβ

cos γ = ctg Α ⋅ ctg Β

i i

1.

cosγ = ctgΑ ⋅ ctgΒ

2.

3.

cos Β = ctgγ ⋅ ctg ( 90° − α ) cos B = ctgγ ⋅ tgα

3.

4.

cos(90° − α ) = ctgB⋅ ctg( 90° − β ) sinα = ctgΒ ⋅ tgβ

4.

5.

cos(90° − β ) = ctgΑ ⋅ ctg( 90° −α )

5.

2.

sinβ = ctgΑ⋅ tgα

6.

cos Α = sin Β ⋅ sin( 90 ° − α ) cosΑ = sinΒ ⋅ cosα

6.

7.

cosγ = sin(90° −α ) ⋅ sin(90° − β ) cosγ = cosα ⋅ cos β

7.

8.

cos B = sin Α ⋅ sin( 90 ° − β ) cos Β = sin Α ⋅ cos β

9.

10.

cos( 90 ° − α ) = sin γ ⋅ sin Α sin α = sin γ ⋅ sin Α sin β = sinγ ⋅ sin Β cos( 90 ° − β ) = sin B ⋅ sin γ

8.

9.

10.

15

LJB=HGHF?LJBYIJ:U HIHJGU?NHJFMEU A:>:Q:
16

17

LZ[ebpZ; KU i i 1. 2.

3.

Ij_^\Zjbl_evgh

cos

HdhgqZl_evgh

θ N = ctg( 90° − α ) ⋅ ctg( 90° − ) 2 2

cos( 90 ° −

θ N ) = ctg ⋅ ctgB 2 2

cos B = ctg 60 ° ⋅ ctg ( 90 ° −

θ ) 2

N θ = tg α ⋅ tg 2 2 N θ sin = ctg ⋅ ctgB 2 2 θ 3 cos B = tg 3 2

cos

4.

cos 60 ° = ctgB ⋅ ctg ( 90 ° − α )

tg α =

1 tgB 2

5.

N cos( 90° − α ) = ctg 60° ⋅ ctg 2 N cos = sin B ⋅ sin 60 ° 2

sin α =

N 3 ctg 3 2

6.

7.

8.

9.

10.

N 3 sin B = 2 2 θ cos( 90° − ) = sin60° ⋅ sin( 90° − α ) sin θ = 3 cos α 2 2 2 N N cos B = sin( 90 ° − α ) ⋅ sin cos B cos sin = ⋅ α 2 2 θ N 1 N θ = sin ⋅ cos cos 60 ° = sin ⋅ sin( 90 ° − ) 2 2 2 2 2 θ θ = ⋅ α sin sin B cos cos( 90° − α ) = sin B ⋅ sin( 90° − ) 2 2 cos

i i 1. 2.

3.

4.

5.

6.

7.

8.

9.

10.

Ihemq_ggu_^_kylvnhjfmefh`gh[uehZgZeh]bqghknhjfmebjh\Zlv \u^_eyy\kljmdlmj_^Zgghc-m]hevghcibjZfb^uijyfhm]hevgucl_ljZw^j BAE 1 F jbk bijbf_gyylhl`_fg_fhgbq_kdbcdjm]

18

IJBF?JUJ?R?GBYA:>:Q Ijbf_j deZkk JZkkfhljbfaZ^Zqmbamq_[gbdZEK:lZgZkygZb ^j>@k
h3 ⋅ 3 1 1 3 2 ⋅ ( 3tg2ϕ − 1). IhwlhfmVibj = ⋅ 3 ⋅ h ⋅ 3 ⋅ ⋅ ( 3tg ϕ − 1) = 4 4 3 h3 ⋅ 3 ⋅ ( 3 tg 2 ϕ − 1 ) . Hl\_l 4 Ijbf_j deZkk J_rbf aZ^Zqm ba k[hjgbdZ ih^ j_^Zdpb_c FB KdZgZ\b >@ k < IjZ\bevgZy lj_m]hevgZy ibjZfb^Z i_j_k_q_gZ iehkdhklvx ijhoh^ys_c q_j_a _z [hdh\h_ j_[jh b \ukhlm < k_q_gbb h[jZah\Zeky lj_m]hevgbd k m]ehf 45º ijb \_jrbg_ ibjZfb^u GZc^bl_m]hef_`^m[hdh\hc]jZgvxbiehkdhklvxhkgh\ZgbyibjZfb^u J?R?GB? Kbgl_lbq_kdbcf_lh^f_lh^hihjguonhjfme < ijbgyluo h[hagZq_gbyo jbk bkihevamy mkeh\b_ b hihjgmx nhjfmem balZ[ebpu;ihemqbfljb]hghf_ljbq_kdmxkbkl_fm o o o  

α + B + 45 = 180 α = 135 − B ⇔  tgB = 2 ⋅ tg( 135 o − B ).  tgB = 2 ⋅ tgα

19

Ba\lhjh]hmjZ\g_gbyihke_^g_ckbkl_fuihemqbf

2 + 2 ⋅ tgB tgB = − 2 ⋅ tg ( 45 o + B ) ⇔ tgB = ⇔ tgB − 1  3 + 17 , tgB = 2  tg B − 3 ⋅ tgB − 2 = 0 2 ⇔ ⇔  tgB ≠ 1  tgB = 3 − 17 ( < 0 ).  2 3 + 17 3 + 17 π ( ≈ 74,3o ). BlZddZd<∈(0; ),lh B = arctg BlZd tgB = 2 2 2 3 + 17 Hl\_l arctg . 2 Ijbf_j deZkk JZkkfhljbf aZ^Zqm ba k[hjgbdZ ih^ j_^Zdpb_c FB KdZgZ\b >@ k < KlhjhgZ hkgh\Zgby ijZ\bevghc lj_m]hevghc ibjZfb^u jZ\gZ Z ;hdh\Zy ]jZgv khklZ\ey_l k iehkdhklvx hkgh\Zgby m]he < GZc^bl_ jZkklhygb_ f_`^m [hdh\uf j_[jhf b g_ i_j_k_dZxs_c_]hklhjhghchkgh\Zgby J?R?GB? :gZeblbdh-kbgl_lbq_kdbcf_lh^f_lh^hihjguonhjfme
a ⋅

3 2

⋅ sin α .

Bkihevamy hihjgmx nhjfmem ba lZ[ebpu ; gZc^_f VLQ.

1 2 1 2 π tg tgB ctg sin = ⋅ ⇒ = ⇒ = ⇒ α α α ( α ∈ ( 0 ; )) : 2 2 tgB 1 ctg + α 2

⇒ sin 2 α =

1 1 sin . ⇒ = α 2 2 1 + 4 ⋅ ctg B 1 + 4 ⋅ ctg B

BlZd d ( SA , BC ) =

Hl\_l

20

a⋅ 2⋅

a⋅

3

2 ⋅ 1 + 4 ⋅ ctg B 2

3

1 + 4 ⋅ ctg 2 B

.

.

Ijbf_jdeZkk J_rbfaZ^Zqmbaihkh[byBNRZju]bgZb<B =hem[_\Z >@ k GZc^bl_ jZ^bmk rZjZ dZkZxs_]hky \k_o j_[_j ijZ\bevghclj_m]hevghcibjZfb^umdhlhjhcklhjhgZhkgh\ZgbyZjZ\gZ Z[hdh\h_j_[jhb jZ\gh J?R?GB? Dhf[bgbjh\Zggucf_lh^ ,H[hkgh\Zgb_bah[jZ`_gbyjbkZ RZjgZau\Z_lkyihem\ibkZgguf \ fgh]h]jZggbd Z fgh]h]jZggbd - ihemhibkZgguf hdheh rZjZ _keb rZj dZkZ_lky\k_oj_[_jfgh]h]jZggbdZ6. >ey ex[hc ijZ\bevghc ibjZfb^u kms_kl\m_l ihem\ibkZgguc rZj ( O ρ ; ρ ) ?]h p_glj e_`bl \ lhqd_ i_j_k_q_gby hkb SO ibjZfb^u k i_ji_g^bdmeyjhf\hkklZ\e_ggufdiehkdhklbijhba\hevghc[hdh\hc]jZgb ba p_gljZ O1 hdjm`ghklb \ibkZgghc \ wlm ]jZgv Lhqdb dZkZgby ( E i ) k j_[jZfb hkgh\Zgby _klv k_j_^bgu wlbo j_[_j Lhqdb dZkZgby ( M i ) k [hdh\ufbj_[jZfbm^Ze_guhl\_jrbghkgh\ZgbygZh^ghblh`_jZkklhygb_ jZ\gh_iheh\bg_j_[jZhkgh\Zgby>hdZ`_fwlh Hq_\b^gh qlh ex[Zy lhqdZ hkb SO ibjZfb^u jZ\ghm^Ze_gZ hl \k_o j_[_j _z hkgh\Zgby gZ jZkklhygb_ jZ\gh_ ^ebg_ hlj_adZ kh_^bgyxs_]h wlm lhqdm k k_j_^bghc E i j_[jZ hkgh\Zgby ih k\hckl\m gZdehgguo bf_xsbo jZ\gu_ijh_dpbb LhqdZ O ρ i_j_k_q_gby emqZ SO k i_ji_g^bdmeyjhf O 1 O ρ , \hkklZ\e_ggufdijhba\hevghc[hdh\hc]jZgbbap_gljZ O 1 \ibkZgghc\g__ hdjm`ghklb h[eZ^Z_l ihfbfh mdZaZggh]h k\hckl\Z ( O ρ E 1 = O ρ E 2 = ) , k\hckl\hf h^bgZdh\hc m^Ze_gghklb hl [hdh\h]h j_[jZ b j_[jZ hkgh\Zgby O ρ E 1 = O ρ M 2 = O ρ M 3 ]^_ E 1 , M 2 , M 3 - lhqdb dZkZgby hdjm`ghklb \ibkZgghc\[hdh\mx]jZgvk_zklhjhgZfb BlZd lhqdZ O ρ jZ\ghm^Ze_gZ hl \k_o j_[_j ijZ\bevghc ibjZfb^u gZ jZkklhygb_ ρ = O ρ E i = O ρ M i Ihwlhfm kms_kl\m_l rZj k p_gljhf O ρ b jZ^bmkhf!ihem\ibkZgguc\ijZ\bevgmxibjZfb^m Ih l_hj_f_ h lj_o i_ji_g^bdmeyjZo jZ^bmku ihem\ibkZggh]h \ ibjZfb^m rZjZ ijh\_^_ggu_ \ lhqdb dZkZgby E i , M i i_ji_g^bdmeyjgu khhl\_lkl\mxsbfj_[jZfibjZfb^uLhqdbdZkZgby M i m^Ze_guhl\_jrbg

A_feydh\:G=_hf_ljby\deZkk_F_lh^j_dhf_g^Zpbbd ij_ih^Z\ZgbxdmjkZ]_hf_ljbbihmq_[ghfmihkh[bx:<Ih]hj_eh\Z– F– k 6

21

22

hkgh\ZgbygZh^ghblh`_jZkklhygb_jZ\gh_iheh\bg_j_[jZhkgh\Zgbyih k\hckl\mdZkZl_evguoijh\_^zgguodrZjmbah^ghclhqdb Ql^ ,,
ρ = O ρ M 2 = SM 2 ⋅ ctgα = ( b −

a=2 a = 2 ⋅ ctgα . ) ⋅ ctgα = 2 b=3

sin

a 1 . = = 2 2 ⋅b 3

Ihhihjghcnhjfme_ balZ[ebpu;

cos α =

2 θ ⋅ sin , 2 3

2 b 3⋅ 3 1 23 − = tgα = 1 aZf_lbfqlh α ≈ 67 ,4° ). 2 2 cos α

hldm^Z cos α

=

Ihwlhfm

ρ = 2 ⋅ ctg α = Hl\_l

4 . 23

4 . 23

23

LJB=HGHF?LJBYIJ:U HIHJGU?NHJFMEU A:>:Q:eyihkljh_gbybah[jZ`_gbyebg_cgh]hm]eZ^ey^\m]jZggh]hm]eZ 2 ijb [hdh\hf j_[j_ SB ijh\_^_f \ ie6$% $) ⊥ SB b kh_^bgbf lhqdm F k lhqdhcCLh]^Zlj_m]hevgbdbAFBbCFBjZ\guihi_j\hfmijbagZdmAF – π −θ π h[sZy:< %K, ∠FBA = ∠FBC == Ihwlhfm∠AFB = ∠AFC = b 2 2 N ∠ $)& N ijbq_f ∠ AFO = ∠ CFO = l d \ jZ\gh[_^j_gghf 2 lj_m]hevgbd_AFCf_^bZgZFOy\ey_lkyb[bkk_dljbkhcm]eZAFCAZf_lbf π qlh N. 2 ,,Ijbf_g_gb_f_lh^Zijyfhm]hevgh]hl_ljZw^jZ < kljmdlmj_ ijZ\bevghc -m]hevghc ibjZfb^u SABCD \u^_ebf ijyfhm]hevguc l_ljZw^j SAOE jbk b d m]eh\uf we_f_glZf _]h lj_o]jZggh]h m]eZ ASEO k _^bgkl\_gguf ijyfuf ^\m]jZgguf m]ehf ijb j_[j_ :H ijbf_gbf ijZ\beh-Ze]hjblf nhjfmebjh\db ^_kylb hihjguo ljb]hghf_ljbq_kdbo nhjfme-k\yahd k bkihevah\Zgb_f fg_fhgbq_kdh]h djm]Zjbk IhemqbflZ[ebpm<

24

25

LZ[ebpZ< KU i i 1.

Ij_^\Zjbl_evgh

cos

HdhgqZl_evgh

θ N = ctg( 90° − α ) ⋅ ctg( 90° − ) 2 2

θ N 2. cos( 90 ° − ) = ctg ⋅ ctgB 2 2 3.

4.

5.

6.

θ cos B = ctg 45° ⋅ ctg( 90° − ) 2

cos 45° = ctgB ⋅ ctg( 90° − α ) N cos( 90° − α ) = ctg45° ⋅ ctg 2 N cos = sin 45 ° ⋅ sin B 2

7.

θ cos( 90° − ) = sin45° ⋅ sin( 90° − α ) 2

8.

cos B = sin( 90 ° − α ) ⋅ sin

9.

10.

N 2 N θ cos 45° = sin ⋅ sin( 90° − ) 2 2

cos( 90° − α ) = sin B ⋅ sin( 90° −

i i

θ ) 2

N θ = tg α ⋅ tg 2 2 N θ sin = ctg ⋅ ctgB 2 2

cos

θ cos B = tg 2

tg α =

3.

5.

N 2

N 2 = sin B 2 2 θ 2 sin = cos α 2 2

cos

cos B = cos α ⋅ sin

2.

4.

2 tgB 2

sin α = ctg

1.

N 2

θ N 2 = sin ⋅ cos 2 2 2 θ sinα = sin B ⋅ cos 2

6.

7.

8.

9.

10.

Ihemq_ggu_ ^_kylv ljb]hghf_ljbq_kdbo nhjfme-k\yahd fh`gh [ueh ZgZeh]bqgh knhjfmebjh\Zlv \u^_eyy \ kljmdlmj_ ^Zgghc ibjZfb^u ijyfhm]hevgucl_ljZw^jABOFjbk

26

IJBF?JUJ?R?GBYA:>:Q Ijbf_j deZkk J_rbf aZ^Zqm ba mq_[gh]h ihkh[by B N RZju]bgZ b < B =hem[_\Z >@ k < ijZ\bevghc q_lujzom]hevghc ibjZfb^_ iehkdbc m]he ijb \_jrbg_ jZ\_g m]em f_`^m [hdh\uf j_[jhf b iehkdhklvx hkgh\Zgby Hij_^_ebl_ ^\m]jZggu_ m]eu f_`^mkhk_^gbfb[hdh\ufb]jZgyfbwlhcibjZfb^u J?R?GB? Kbgl_lbq_kdbcf_lh^f_lh^hihjguonhjfme Bkihevamy mkeh\b_ b hihjgu_ nhjfmeu b ba lZ[ebpu < ihemqbfljb]hghf_ljbq_kdmxkbkl_fm   α =θ  θ  cos α = 2 ⋅ sin 2   sin Φ ⋅ cos θ = 2  2 2 2 Ba\lhjh]hmjZ\g_gbykbkl_fuihke_aZf_gu .ihemqbf α α α π 2 ⋅ sin 2 + 2 ⋅ sin − 1 = 0 hldm^Zld ∈ ( 0 ; ) , 2 2 2 4 5 −1 α sin = bihwlhfm 2 2⋅ 2

α cos = 2

α 1 − sin 2 = 2

 5 −1  1 −   2 ⋅ 2  

2

=

1+ 5 2

Balj_lv_]hmjZ\g_gbykbkl_fuihke_aZf_gu .ihemqbf Φ 2 Φ 1+ 5 2 sin ⋅ = ⇔ sin = ⇔ 2 2 2 2 1+ 5

⇔ Φ = 2 ⋅ arcsin Hl\_lΦ = 2 ⋅ arcsin

2 1+ 5

2 1+ 5

ld

Φ π π ∈( ; ). 2 4 2

( ≈ 103 ,66 O ) .

AZf_qZgb_
Φ = 2 − 5 ihemqbf ^jm]mx 2

nhjfmhl\_lZΦ = arccos( 2 − 5 )( ≈ 103 ,66 O ) . 27

Ijbf_j deZkk JZkkfhljbf aZ^Zqm ba k[hjgbdZ ih^ j_^Zdpb_c FB KdZgZ\b >@ k < JZkklhygby hl p_gljZ hkgh\Zgby ijZ\bevghc q_luj_om]hevghc ibjZfb^u ^h [hdh\hc ]jZgb b ^h [hdh\h]h j_[jZjZ\gukhhl\_lkl\_ggha b b GZc^bl_^\m]jZggucm]heijbhkgh\Zgbb ibjZfb^u J?R?GB? Dhf[bgbjh\Zggucf_lh^ ,H[hkgh\Zgb_bah[jZ`_gby
 a  1 b a2 =  − 1= 2 −1  2  cos B cos α ⇔  cos 2 α b ⋅cos B ⇒ 2 2 tgB = 2 ⋅ tgα  α = ⋅ tg B 2 tg  2 ⋅( a 2 − b 2 ⋅cos 2 B ) 2 2 2 2 2 2 ⇒ tg B = ⇔ ⋅ = ⋅ − ⋅ b sin B 2 ( a b cos B )⇔ 2 2 b cos B 2 ⋅a 2 − b 2 2 ⋅a 2 − b 2 2 ⇔ cos B = < 1. , ]^_ 0 < 2 2 b b Hl\_l :

28

2 ⋅a 2 − b 2 B = arccos , b

]^_

a < b < a ⋅ 2.

kihkh[Dlhc`_ljb]hghf_ljbq_kdhckbkl_f_fuijb^zf_keb mjZ\gy_f^ebgmhlj_adZH<\ujZ`_ggmx^\mfykihkh[Zfb b .
 a⋅ 2 b  a b =   =  sin B sin α ⇔  cos B cos α tgB = 2 ⋅ tgα tgB = 2 ⋅ tgα 

b l^kf 1 kihkh[

kihkh[ < _]h hkgh\_ khklZ\e_gb_ ljb]hghf_ljbq_kdhc kbkl_fu \dexqZxs_c \ k_[y hihjgmx nhjfmem ba lZ[ebpu < b mjZ\g_gb_ khklZ\e_ggh_ gZ hkgh\_ f_lh^Z mjZ\gb\Zgby ^ey ^ebgu hlj_adZ H: H', \ujZ`_gghc^\mfyjZagufbkihkh[Zfb Φ
 2 ⋅a 2 Φ 2  Φ a ⋅ 2  tg = 2 b tg ⋅ = 2  b ⋅ sin 2 B 2  2 sin B ⇔ ⇒ −1 = 2 1   Φ sin B 2 1 1 sin B  2 Φ − = sin 2 B −  cos =  2 2  cos 2  2 ⋅a 2 2 ⋅b 2 − 2 ⋅a 2 2 , ]^_ a < b < a ⋅ 2 . = 2 ⇔ sin B = 2 2 b ⋅ sin B b 2 ⋅ b 2 − 2 ⋅a 2 Hl\_l : B = arcsin b

, ]^_

a < b < a⋅ 2.

29

LJB=HGHF?LJBYIJ:U HIHJGU?NHJFMEU A:>:Q:ey ihkljh_gby ijh_dpbhggh]h bah[jZ`_gbyijZ\bevgh]h-m]hevgbdZ A1 A2 A3 A4 A5 e_`Zs_]h\hkgh\Zgbb ^Zgghc ibjZfb^u k \_jrbghc S [m^_f hibjZlvky gZ ke_^mxsb_ ljb _]h ieZgbf_ljbq_kdbok\hckl\Z – dZ`^Zy_]h^bZ]hgZeviZjZee_evgZh^ghcbaklhjhg – dZ`^Zy _]h ^bZ]hgZev ^_eblky ^jm]hc ^bZ]hgZevx \ djZcg_f b 5 +1 ≈ 1 ,6 ; kj_^g_fhlghr_gbbjZ\gufqbkemNb^by 2 – _]h p_glj e_`bl \ lhqd_ i_j_k_q_gby ^\mo f_^bZg ^\mo jZ\gh[_^j_gguo lj_m]hevgbdh\ khhl\_lkl\_ggh A1 A4 A2 b A2 A5 A3 , ijh\_^_gguodbohkgh\Zgbyf A1 A2 b A2 A3 . Ihwlhfm ihkljh_gb_ ijh_dpbhggh]h bah[jZ`_gby ijZ\bevgh]h 5-m]hevgbdZfh`ghhkms_kl\blvke_^mxsbfh[jZahf Kljhbfijhba\hevguclj_m]hevgbd A1 A4 A2 \dhlhjhf? –k_j_^bgZ klhjhgu A1 A2 , A4 E –f_^bZgZ Hlj_adb A4 A1 b A4 A2 ^_ebf lhqdZfb L b M khhl\_lkl\_ggh \ A L A M 5 djZcg_f b kj_^g_f hlghr_gbb ijb[eb`_ggh iheZ]Zy qlh 4 = 4 ≈ . LA1 MA2 3 AZf_lbfqlhLM|| A1 A2 . Ijh\h^bf A1 A5 iZjZee_evgh A2 A4 ^h i_j_k_q_gby \ lhqd_ A5 k ijh^he`_gb_f A2 L aZlhqdmL. Ijh\h^bf A2 A3 iZjZee_evgh A1 A4 ^h i_j_k_q_gby \ lhqd_ A3 k ijh^he`_gb_f A1 M aZlhqdmF. Kljhbfhlj_adb A3 A4 b A4 A5 .

30

31

P_glj O ihemq_ggh]h bah[jZ`_gby ijZ\bevgh]h -m]hevgbdZ A1 A2 A3 A4 A5 hij_^_ey_f dZd lhqdm i_j_k_q_gby f_^bZg lj_m]hevgbdh\ A1 A4 A2 , A2 A5 A3 ijh\_^_gguokhhl\_lkl\_gghdklhjhgZf A1 A2 b A2 A3 . π Lh]^Z SO – \ukhlZ ^Zgghc ibjZfb^u ∠ SA1 O = α ]^_ 0 < α < ; 2 π θ 2π ∠ SEO =B]^_0 < B < ; ∠ A1 SE = ]^_0 < θ < . 2 2 5 Kp_evxihkljh_gbybah[jZ`_gbyebg_cgh]hm]eZ^ey^\m]jZggh]hm]eZ ijb[hdh\hfj_[j_ SA2 ijh\_^_f\ie A1 SA2 ) A1 F⊥SA2 bkh_^bgbflhqdmF N l d k lhqdhc A3 Lh]^Z ∠ A1 FA3 N ijbq_f ∠A1 FN = ∠ A3 FN = 2 3π Nπ. f_^bZgZFNy\ey_lky[bkk_dljbkhcm]eZ A1 FA3 AZf_lbfqlh 5 II. Ijbf_g_gb_f_lh^Zijyfhm]hevgh]hl_ljZw^jZ < kljmdlmj_ ijZ\bevghc -m]hevghc ibjZfb^u SA1 A2 A3 A4 A5 \u^_ebf ijyfhm]hevguc l_ljZw^j SA1 OE jbk b d m]eh\uf we_f_glZf _]h lj_o]jZggh]h m]eZ A1 SOE k _^bgkl\_gguf ijyfuf m]ehf ijb j_[j_ A1 O ijbf_gbf ijZ\beh-Ze]hjblf nhjfmebjh\db ^_kylb hihjguo ljb]hghf_ljbq_kdbo nhjfme-k\yahd k bkihevah\Zgb_f fg_fhgbq_kdh]h djm]Zjbk IhemqbflZ[ebpm= LZ[ebpZ= KU i i

HdhgqZl_evgh

N N θ θ = ctg( 90° − α ) ⋅ ctg( 90° − ) cos = tg α ⋅ tg 2 2 2 2 N θ N θ sin = ctg ⋅ ctgB cos( 90 ° − ) = ctg ⋅ ctgB 2 2 2 2 θ θ cos B ctg 36 tg = ° ⋅ cos B = ctg 36 ° ⋅ ctg ( 90 ° − ) 2 2

1. cos

1.

2.

2.

3. 4. 5.

32

Ij_^\Zjbl_evgh

i i

cos 36 ° = ctgB ⋅ ctg ( 90 ° − α ) cos( 90 ° − α ) = ctg 36 ° ⋅ ctg

N 2

tg α = cos 36 ° ⋅ tgB N sin α = ctg 36 ° ⋅ ctg 2

3. 4. 5.

i i

Ij_^\Zjbl_evgh

i i

HdhgqZl_evgh

N = sin 36° ⋅ sin B 2

cos

N = sin 36° ⋅ sin B 2

6.

θ ) = sin 36° ⋅ sin( 90° − α ) 2

θ sin = sin 36° ⋅ cosα 2

7.

N 2

8.

N θ ⋅ cos 2 2

9.

θ ) sin α = sin B ⋅ cos θ 2 2

10.

6.

cos

7.

cos( 90° −

8.

N cos B = sin( 90 ° − α ) ⋅ sin 2

9.

cos 36 ° = sin

10.

cos( 90° − α ) = sin B ⋅ sin( 90° −

θ N ⋅ sin( 90 ° − ) 2 2

cos B = cos α ⋅ sin cos 36° = sin

Ihemq_ggu_ ^_kylv ljb]hghf_ljbq_kdbo nhjfme fh`gh [ueh ZgZeh]bqgh knhjfmebjh\Zlv \u^_eyy \ kljmdlmj_ ^Zgghc -m]hevghc ibjZfb^uijyfhm]hevgucl_ljZw^j A1 A2 NF jbk Ihe_aghihfgblvf_lh^\uqbke_gbyke_^mxsboagZq_gbc7:

5 − 1 10 + 2 5 sin 18 ° = cos 18° = 4 4 5 + 1 10 − 2 5 ° = cos 36 sin 36° = 4 4 :e]_[jZ ^ey deZkkZ / G Y
33

IJBF?JUJ?R?GBYA:>:Q Ijbf_j deZkk GZc^bl_ m]eu khklZ\ey_fu_ k hkgh\Zgb_f [hdh\uf j_[jhf b [hdh\hc ]jZgvx ijZ\bevghc iylbm]hevghc ibjZfb^u m dhlhjhc[hdh\u_]jZgb–jZ\ghklhjhggb_lj_m]hevgbdb. J?R?GB? Kbgl_lbq_kdbcf_lh^f_lh^hihjguonhjfme
θ sin = sin 36° ⋅ cosα hldm^Z Ihhihjghcnhjfme_ balZ[ebpu= 2 5+ 5 . 1 4 cos α = = = 10 2 ⋅ sin 36 ° 2 ⋅ 10 − 2 ⋅ 5 Ihhihjghcnhjfme_ balZ[ebpu= cos Β = ctg 36° ⋅ tg

cos B =

=

5 +1 3 ⋅ 10 − 2 ⋅ 5

( 5 + 1 )3 2 ⋅ 2 ⋅ 3⋅ 5

Hl\_l . = arccos

=

=

5 +1 3 ⋅ 2⋅ 5 ⋅

8 ⋅( 5 + 2 ) 2 ⋅ 2 ⋅ 3⋅ 5

=

5+ 5 ( ≈ 31 ,7 ° ); B = arccos 10

5 −1

5 +2 = 3⋅ 5

θ hldm^Z 2

=

5 + 2⋅ 5 . 15

5 + 2⋅ 5 ( ≈ 37 ,4 ° ). 15

AZf_qZgb_:gZeh]bqgmxaZ^Zqmfh`ghknhjfmebjh\Zlvebrv^eyn=3; 4. Ijbf_j deZkk RZj \ibkZgguc \ ijZ\bevgmx iylbm]hevgmx ibjZfb^m b rZj hibkZgguc hdheh wlhc ibjZfb^u bf_xl h[sbc p_glj >hdZaZlv qlh hdjm`ghklb hibkZggu_ hdheh hkgh\Zgby b [hdh\uo ]jZg_c ibjZfb^u jZ\gu GZc^bl_ m]eu khklZ\ey_fu_ k hkgh\Zgb_f [hdh\uf j_[jhfb[hdh\hc]jZgvxZlZd`_^\m]jZggucm]heijb[hdh\hfj_[j_ J?R?GB? Dhf[bgbjh\Zggucf_lh^ ,H[hkgh\Zgb_bah[jZ`_gbyBa\_klgu^\ZnZdlZ8.
34

SO ibjZfb^u k k_j_^bgguf i_ji_g^bdmeyjhf d [hdh\hfm j_[jm gZijbf_j SA1 ijh\_^zgguf\ ie .( SAO 1 ) . Ih mkeh\bx p_glj O r \ibkZgghc b p_glj O R hibkZgghc kn_j kh\iZ^ZxlH[hagZqbfh[sbcp_gljwlbokn_j, ,,>hdZaZl_evkl\hkbgl_lbq_kdbcf_lh^ LZd dZd IS = IA1 = IA2 = R dZd jZ^bmku hibkZggh]h rZjZ lh O 1 S = O 1 A1 = O 1 A2 dZd ijh_dpbb gZ [hdh\mx ]jZgv SA1 A2 jZ\guo gZdehgguo Ihwlhfm O1 – p_glj hdjm`ghklb hibkZgghc hdheh [hdh\hc ]jZgb Ijyfhm]hevgu_ lj_m]hevgbdb IOA1 b IO 1 A1 jZ\gu ih dZl_lm b ]bihl_gma_ IO = IO 1 = r , IA1 – h[sZy Ihwlhfm OA1 = O 1 A1 l_ hdjm`ghklb hibkZggu_ hdheh hkgh\Zgby b [hdh\uo ]jZg_c ibjZfb^u jZ\guQl^ ,,,
1 = 2 ⋅ cos 18°

2 10 + 2 ⋅ 5

=

θ = sin 36 ° ⋅ cos α ihemqbf qlh 2

5− 5 . 10

Bkihevamy hihjgmx nhjfmem ba lZ[ebpu = cos Β = ctg 36° ⋅ tg ihemqbfqlh cos Β =

tg 18° cos 36° 5 +1 5 = = = tg 36° 1 + cos 36° 5 5 +5

Bkihevamy hihjgmx nhjfmem ba lZ[ebpu = cos Β = cos α ⋅ sin

θ , 2

Φ , 2

Φ Φ cos Β 5+ 5 5− 5 = = hldm^Z cos = 2 cos α 10 2 10 Φ LZdbf h[jZahf ^ey ^Zgghc ibjZfb^u α = ^ey^jm]boijZ\bevguo

ihemqbfqlh sin

ibjZfb^wlhg_\_jgh Φ Hl\_l α = = arccos 2

2

5 5− 5 ( ≈ 63.4° ) . ( ≈ 58 .3 ° ) ; Β = arccos 5 10

35

LJB=HGHF?LJBYIJ:U HIHJGU?NHJFMEU A:>:Q:ey ihkljh_gby ijh_dpbhggh]h bah[jZ`_gby ijZ\bevgh]h -m]hevgbdZ A1 A2 A3 A4 A5 A6 e_`Zs_]h \ hkgh\Zgbb^ZgghcibjZfb^uk\_jrbghcSfh`ghihkljhblvijhba\hevguc iZjZee_eh]jZff A1 A2 A3 O b gZ emqZo A1 O , A2 O , A3 O hleh`blv hlj_adb OA4 = OA1 , OA5 = OA2 , OA6 = OA3 < j_amevlZl_ ihemqbf bah[jZ`_gb_ ijZ\bevgh]h-m]hevgbdZkp_gljhfHLh]^ZSO –\ukhlZ^ZgghcibjZfb^u π ∠ SA1 O = α]^_0 < α < Imklv?k_j_^bgZj_[jZ A1 A2 , lh]^Z ∠ SEO =B, 2 π θ π ]^_0 < B < ; ∠ A1 SE = ]^_0 < θ < Kp_evxihkljh_gbybah[jZ`_gby 2 2 3 ebg_cgh]h m]eZ ^ey ^\m]jZggh]h m]eZ ijb [hdh\hf j_[j_ SA6 ijh\_^_f \ ie SA1 A6 ) A1 F⊥SA6 bkh_^bgbflhqdmFklhqdhc A5 Lh]^Z ∠ A1 FA5 N, N ijbq_f ∠ A1 FN = ∠ A5 FN = l d \ jZ\gh[_^j_gghf lj_m]hevgbd_ 2 A1 FA5 f_^bZgZ FN y\ey_lky [bkk_dljbkhc m]eZ A1 FA5 AZf_lbf qlh 2π <Φ <π . 3 II. Ijbf_g_gb_f_lh^Zijyfhm]hevgh]hl_ljZw^jZ < kljmdlmj_ ijZ\bevghc -m]hevghc ibjZfb^u SA1 A2 A3 A4 A5 A6 \u^_ebf ijyfhm]hevguc l_ljZw^j SA1 OE jbk b d m]eh\uf we_f_glZf _]h lj_o]jZggh]h m]eZ A1 SOE k _^bgkl\_gguf ijyfuf m]ehf ijb j_[j_ A1 O ijbf_gbf ijZ\beh-Ze]hjblf nhjfmebjh\db ^_kylb hihjguo ljb]hghf_ljbq_kdbo nhjfme-k\yahd k bkihevah\Zgb_f fg_fhgbq_kdh]h djm]Zjbk IhemqbflZ[ebpm>

36

37

LZ[ebpZ> KU i i 1.

2.

Ij_^\Zjbl_evgh

cos

HdhgqZl_evgh

θ N = ctg ( 90 ° − α ) ⋅ ctg ( 90 ° − ) 2 2 θ N cos( 90 ° − ) = ctg ⋅ ctgB 2 2

θ ) 2

3.

cos B = ctg 30 ° ⋅ ctg ( 90 ° −

4.

cos 30 ° = ctgB ⋅ ctg ( 90 ° − α )

5. 6.

cos( 90 ° − α ) = ctg 30 ° ⋅ ctg

N cos = sin 30 ° ⋅ sin B 2

i i

N 2

N θ = tg α ⋅ tg 2 2 N θ sin = ctg ⋅ ctgB 2 2 cos

θ 2

cos B =

3 tg

tg α =

3 tgB 2

1.

2. 3.

4.

N sin α = 3ctg 2

5.

cos

6.

N 1 = sin B 2 2

θ 1 θ sin = cos α 7. cos( 90° − ) = sin 30° ⋅ sin( 90° − α ) 2 2 2 N N α = ⋅ cos B cos sin cos B = sin( 90° − α ) ⋅ sin 8. 2 2 N θ N 3 9. θ cos 30° = sin ⋅ sin( 90° − ) = sin ⋅ cos 2 2 2 2 2 θ θ 10. cos( 90 ° − α ) = sin B ⋅ sin( 90° − ) sinα = sin B ⋅ cos 2 2

7.

8.

9. 10.

Ihemq_ggu_ ^_kylv ljb]hghf_ljbq_kdbo nhjfme-k\yahd fh`gh [ueh ZgZeh]bqgh knhjfmebjh\Zlv \u^_eyy \ kljmdlmj_ ^Zgghc -m]hevghc ibjZfb^uijyfhm]hevgucl_ljZw^j A1 A6 NF jbk .

38

IJBF?JUJ?R?GBYA:>:Q Ijbf_j deZkk J_rbf aZ^Zqm ba mq_[gh]h ihkh[by BN RZju]bgZb<B=hem[_\Z>@k fh`_lkh\iZ^Zlvk 4 π p_gljhf H hkgh\Zgby ibjZfb^u _keb α = fh`_l gZoh^blvky \gmljb 4 π ibjZfb^u_keb α < gZijh^he`_gbbSOaZlhqdmH. 4 <^ZgghcaZ^Zq_p_glj O R hibkZgghckn_jufh`_leb[he_`Zlv\gmljb hlj_adZSOkemqZc eb[hkh\iZ^Zlvkp_gljhfHhkgh\ZgbykemqZc ,, tgα = 2 w\jbklbq_kdbf ijbgpbihf iZjZ^b]fu wlZ nhjfmeZ [m^_l ih^kdZau\Zlv ^Zevg_crbcoh^fuke_c JZkkfhljbfk_q_gb_ibjZfb^uiehkdhklvx ( SA1 O ) ijhoh^ys_cq_j_a \ukhlmSOb[hdh\h_j_[jh SA1 jbkZ

39

kemqZc

S S

R R

α

K

OR

OR

L

Or

R

O

r

Or

r

α A1

B r B

B

E

A4

O

JbkZ Jbk[ < ijyfhm]hevghf ∆ O R A1 O : A1 O = A1 O R 2 − O R O 2 = R 2 − 4 r 2 ]^_ R>2r.
SO = A1 O

R + 2r R − 4r 2

2

.

JZkkfhljbfk_q_gb_ibjZfb^uiehkdhklvx(SOE)ijhoh^ys_cq_j_a_z \ukhlmSObZihn_fmSEjbk[ ( R + r )2 − r 2 SL = .
R bf__fmjZ\g_gb_dhlhjh_ihke_\ha\_^_gbyh[_bo r

qZkl_c\d\Z^jZlk\_^zfdd\Z^jZlghfm 2

R + 2r R 2 − 4r 2

40

R   + 2 2 2 3 R + 2 Rr 3  R  R r   = ⋅ ⇔ = ⋅   + 2 ⋅  . 2 2 r 4  r  r  R     −4 r

Ihke_aZf_gu

R = t ]^_t>2ihemqbfmjZ\g_gb_ r

t>2 1 3 3 t>2 2 = ⋅ t ⋅ ( t + 2 )⇔ = ⋅ t ⇔ 3t − 6 t − 4 = 0 ⇔ 2 4 t−2 4 t −4

( t + 2 )2

 3 + 21 > 2, t = 3 ⇔   3 − 21 < 0 , ihwlhfm g_ijb]h^gh . t = 3  R 3 + 21 = . r 3

BlZd^eyi_j\h]hkemqZyihemqZ_fhl\_l

π . 4 Ih-ij_`g_fm \ dZq_kl\_ hkgh\u ^ey khklZ\e_gby mjZ\g_gby hlghkbl_evgh R hlghr_gby \havf_f nhjfmem ba lZ[ebpu > JZkkfhljbf k_q_gb_ r ibjZfb^uiehkdhklvxijhoh^ys_cq_j_a\ukhlmSObZihn_fmSEjbk\ < ijyfhm]hevghf∆ SO r L ( ∠SLO r = 90° , ∠SO r L = Β ): kemqZcLhqdZ H R kh\iZ^Z_lkp_gljhfHhkgh\Zgbyihwlhfm α =

2

( R − r )2 − r 2 SL tgΒ = = = LO r r Bf__f^ey\uqbke_gby

R 2 − 2 Rr R  R =   − 2 ⋅ ]^_R>2r. r r r R r

mjZ\g_gb_dhlhjh_k\_^_f dd\Z^jZlghfmkemqZc

S

2

3  R R R ⋅   − 2 ⋅ = 1 ]^_ > 2 . 2 r r r R Ihke_aZf_gu = t ]^_t>2, r 2 ihemqbf 3t − 6 t − 4 = 0 , R 3 + 21 hldm^Z t = = . B r 3 E 3 + 21 . Hl\_l 3

L r

B r

R Or

B O = OR

Jbk\

41

LJB=HGHF?LJBYIJ:
n-M=HEVGHCIBJ:FB>U HIHJGU?NHJFMEU A:>:Q:
42

43

ljb]hghf_ljbq_kdbo nhjfme-k\yahd k bkihevah\Zgb_f fg_fhgbq_kdh]h djm]Zjbk IhemqbflZ[ebpm? LZ[ebpZ?

i i 1. 2.

3.

4. 5.

6. 7.

8. 9.

10.

KU Ij_^\Zjbl_evgh

θ N = ctg( 90° − α ) ⋅ ctg( 90° − ) 2 2 θ N cos( 90 ° − ) = ctg ⋅ ctgB 2 2

cos

cos B = ctg

θ 180 ° ⋅ ctg ( 90° − ) n 2

HdhgqZl_evgh

cos sin

N θ = tg α ⋅ tg 2 2

θ N = ctg ⋅ ctgB 2 2

cos B = ctg

1. 2.

θ 3. 180 ° ⋅ tg n 2

180 ° 180 ° = ctgB ⋅ ctg ( 90 ° − α ) = tg α ⋅ ctgB cos n n N 180 ° 180° N sin α = ctg ctg ⋅ ctg cos( 90° − α ) = ctg n 2 n 2 N 180 ° N 180 ° ⋅ sin B cos = sin cos = sin ⋅ sin B 2 n 2 n θ 180 ° 180° θ ⋅ cos α cos( 90° − ) = sin ⋅ sin( 90° − α ) sin = sin 2 n 2 n N N cos B = cosα ⋅ sin cos B = sin( 90 ° − α ) ⋅ sin 2 2 N θ 180 ° 180 ° N θ cos = sin ⋅ sin( 90 ° − ) cos n = sin 2 ⋅ cos 2 n 2 2 θ θ = ⋅ α sin sin B cos cos( 90° − α ) = sin B ⋅ sin( 90° − ) 2 2 cos

i i

4. 5.

6. 7.

8. 9.

10.

Ihemq_ggu_ ^_kylv ljb]hghf_ljbq_kdbo nhjfme-k\yahd fh`gh [ueh ZgZeh]bqgh knhjfmebjh\Zlv \u^_eyy \ kljmdlmj_ ^Zgghc Q-m]hevghc ibjZfb^uijyfhm]hevgucl_ljZw^j A1 A2 NF jbk NhjfmeubalZ[ebpu?fh`ghjZa^_eblvgZ]jmiiu

44

N 2 Bo^hdZaZl_evkl\hm^h[ghijh\_klbf_lh^hf\hkoh^ys_]hZgZebaZgZhkgh\_ j_r_gby ijyfhm]hevguo lj_m]hevgbdh\ \oh^ysbo \ kljmdlmjm SA1 OE jbk Ijb\_^_f ijbf_j ijyfhm]hevgh]h l_ljZw^jZ ^hdZaZl_evkl\Znhjfmeu 180° OE OE EA1 θ ⋅ tg ⇐ = ⋅ qlhbklbggh (3) cos B = ctg n 2 SE EA1 SE N gh g_ ,, ]jmiiZ – wlh nhjfmeu kh^_j`Zsb_ \_ebqbgm m]eZ 2 kh^_j`Zsb_< Bo^hdZaZl_evkl\hih-ij_`g_fmm^h[ghijh\_klb f_lh^hf \hkoh^ys_]h ZgZebaZ gh gZ hkgh\_ j_r_gby ijyfhm]hevguo lj_m]hevgbdh\ \oh^ysbo \ kljmdlmjm ^jm]h]h ijyfhm]hevgh]h l_ljZw^jZ A1 A2 NF jbk Ijb\_^_fijbf_j^hdZaZl_evkl\Znhjfmeu ,]jmiiZ–wlhnhjfmeug_kh^_j`Zsb_\_ebqbgmm]eZ

(1) cos

N FN FN FA 2 θ = tg α ⋅ tg ⇐ = ⋅ qlhbklbggh 2 2 FA 1 FA 2 FA 1

N b< 2 Bo ^hdZaZl_evkl\h wdhghfg__ \k_]h ijh\_klb Ze]_[jZbq_kdbf f_lh^hf bkdexq_gby \_ebqbg g_dhlhjuo m]eh\ ba jZg__ ^hdZaZgguo nhjfme Ijb\_^_fijbf_ju NhjfmeZ \u\h^blky ba ^hdZaZgguo nhjfme b iml_f θ bkdexq_gby\_ebqbgum]eZ \k_m]euhklju_ 2 180 °  θ tg cos B tg = ⋅  180 ° θ  2 n cos B ctg tg = ⋅    n 2 N ⇔  ⇒  sin 1 180 N ° θ 2  cos  = = sin ⋅ cos  180 ° θ  n 2 2 cos cos  2 n N sin2 180° 2 ⇔ cos2 180° + ( 1 − sin2 B)⋅ sin2 180° = sin2 N ⇔ ⇒ 1 + cos2 B ⋅ tg2 = 180° n n n 2 cos2 n 180° 180 ° N 2 N = sin 2 B ⋅ sin 2 ⇔ ⇔ 1 − sin 2 B ⋅ sin 2 = sin 2 ⇔ cos 2 n n 2 III]jmiiZ–wlhnhjfmeukh^_j`Zsb_\_ebqbgum]eh\

⇔ cos

N 180 ° = sin B ⋅ sin . 2 n

45

NhjfmeZ \u\h^blky ZgZeh]bqgh ba nhjfme b f_lh^hf θ bkdexq_gby\_ebqbgum]eZ . 2 NhjfmeZ \u\h^blky ba nhjfme b f_lh^hf bkdexq_gby \_ebqbgum]eZαgZhkgh\_nhjfmeu

180 ° 180 °   tg ctgB ctgB cos cos α = ⋅ ctg α = ⋅    n n ⇒ ⇔   ° ° N 180 180 N  ctg  sin α = ctg = tg ⋅ sin α ⋅ ctg   2 n n 2 N 180 ° N θ ⇒ ctg ⋅ ctgB = sin ⋅ cos α ⇔ ctg ⋅ ctgB = sin . 2 2 2 n

IJBF?JUJ?R?GBYA:>:Q Ijbf_j deZkk J_rbf aZ^Zqm ba k[hjgbdZ ih^ j_^Zdpb_c FB KdZgZ\b >@ k ; Hlghr_gb_ iheghc ih\_joghklb ijZ\bevghc Q-m]hevghc ibjZfb^u d iehsZ^b hkgh\Zgby jZ\gh l GZc^bl_ m]hef_`^m[hdh\ufj_[jhfbiehkdhklvxhkgh\Zgby J?R?GB? Kbgl_lbq_kdbcf_lh^f_lh^hihjguonhjfme
S hkg  1  + S hkg = S hkg ⋅  + 1 ⇒ cos B  cos B  1 S 1 , ⇒ = + 1 = l ⇒ cos B = l −1 S hkg cos B

_klv S = S [ + S hkg =

]^_^ey B ∈ ( 0 ;

π ) cos B ∈ ( 0 ;1 ) bihwlhfml>2. 2

Ihhihjghcnhjfme_ balZ[ebpu? tgα = cos LZddZd^eyl >2 tgB = lh tg α = cos π ⋅

n

Hl\_lα

46

π ⋅ tgΒ . n

1 − 1 = (l- 1 ) 2 − 1 = l ⋅ (l- 2 ) , 2 cos B π  .

l ⋅ ( l − 2 ) b α = arctg  cos ⋅ l ⋅ ( l − 2 )  n  

π   = arctg  cos ⋅ l ⋅ ( l − 2 )  ]^_l > 2. n  

Ijbf_jdeZkk @k
1) LZd dZd Vibj =

a a2 n⋅ a2 = Ihwlhfm S hkg = . 2) S ∆ A1OA2 180° 180° 180° 2 tg 4 tg 4 tg n n n a ⋅ tgα Ba¨ SOA2 : H = OA2 ⋅ tgα = ]^_ α = ∠SA2 O . 180° 2 sin n θ 180° ⋅ cos α ihwlhfm Ihhihjghcnhjfme_ balZ[ebpu? sin = sin 2 n θ 180° − sin 2 sin 2 sin α n 2 = , tgα = θ cos α sin 2 θ 180° θ 180° ]^_ sin ≤ 60° ijb n ≥ 3 . > sin > 0 ld 0 < < n 2 2 n θ 180° − sin 2 a ⋅ sin 2 n 2 . BlZd H ibj = θ 180° ⋅ sin 2 sin n 2 1 1 = ⋅ a ⋅ OE = ⋅ a ⋅ 2 2

θ n ⋅ a 3 ⋅ sin 180° − θ  ⋅ sin 180° + θ  180°     − sin 2 2 2  n  n n 2 Hl\_lV = . = θ θ 180° 180° 180° 180° ⋅ tg ⋅ sin ⋅ tg ⋅ sin 24 sin 24 sin n n 2 n n 2 AZf_qZgb_ J_rzggZy aZ^ZqZ y\ey_lky l_hj_lbq_kdbf h[h[s_gb_f aZ^Zqbbalh]h`_mq_[gbdZEK:lZgZkygZb^j>@k n ⋅ a 3 ⋅ sin 2

47

§ 8. A:>:QB>EYK:FHKLHYL?EVGUOBDHGLJHEVGUOJ:;HL <BDE:KK:O AZ^ZqZ 1.
48

AZ^ZqZKlhjhgZhkgh\ZgbyijZ\bevghcn-m]hevghcibjZfb^ujZ\gZ Z[hdh\h_j_[jhjZ\ghGZclb Z m]eu khklZ\ey_fu_ k hkgh\Zgb_f [hdh\uf j_[jhf b [hdh\hc ]jZgvx iehkdbc m]he ijb \_jrbg_ Z lZd`_ ^\m]jZgguc m]he f_`^m khk_^gbfb[hdh\ufb]jZgyfbwlhcibjZfb^udeZkk [ iehsZ^viheghcih\_joghklbibjZfb^udeZkk \ h[tzfibjZfb^udeZkk ] h[tzf rZjZ hibkZggh]h hdheh ibjZfb^u b iehsZ^v _]h kn_ju (11 deZkk ^ h[tzf\ibkZggh]h\ibjZfb^mrZjZbiehsZ^v_]hkn_judeZkk _ h[tzf ihem\ibkZggh]h \ ibjZfb^m rZjZ b iehsZ^v _]h kn_ju (11 deZkk AZ^ZqZ < ijZ\bevghc n-m]hevghc ibjZfb^_ iehkdbc m]he ijb \_jrbg_ jZ\_g m]em f_`^m [hdh\uf j_[jhf b iehkdhklvx hkgh\Zgby GZc^bl_ ^\m]jZggu_ m]eu khklZ\ey_fu_ [hdh\hc ]jZgvx k hkgh\Zgb_f b khk_^gbfb[hdh\ufb]jZgyfbdeZkk AZ^ZqZ JZkklhygby hl p_gljZ hkgh\Zgby ijZ\bevghc n-m]hevghc ibjZfb^u^h[hdh\hc]jZgbb^h[hdh\h]hj_[jZjZ\gukhhl\_lkl\_gghZbb. GZc^bl_ Z m]eu khklZ\ey_fu_ k hkgh\Zgb_f [hdh\uf j_[jhf b [hdh\hc ]jZgvx iehkdbc m]he ijb \_jrbg_ Z lZd`_ ^\m]jZgguc m]he f_`^m khk_^gbfb[hdh\ufb]jZgyfbwlhcibjZfb^udeZkk [ iehsZ^viheghcih\_joghklbibjZfb^udeZkk \ h[tzfibjZfb^udeZkk ] h[tzf rZjZ hibkZggh]h hdheh ibjZfb^u b iehsZ^v _]h kn_ju (11 deZkk ^ h[tzf\ibkZggh]h\ibjZfb^mrZjZbiehsZ^v_]hkn_judeZkk _ h[tzfihem\ibkZggh]h\ibjZfb^mrZjZb_]hkn_judeZkk AZ^ZqZ
49

AZ^ZqZ < ijZ\bevghc n-m]hevghc ibjZfb^_ p_glj hibkZgghc kn_ju e_`blgZih\_joghklb\ibkZgghcGZc^bl_ Z hlghr_gbyjZ^bmkh\hibkZgghcb\ibkZgghckn_jdeZkk [ m]eukhklZ\ey_fu_hkgh\Zgb_f[hdh\ufj_[jhfb[hdh\hc]jZgvx iehkdbc m]he ijb \_jrbg_ Z lZd`_ ^\m]jZgguc m]he f_`^m khk_^gbfb [hdh\ufb]jZgyfbwlhcibjZfb^udeZkk AZ^ZqZ < ijZ\bevghc n-m]hevghc ibjZfb^_ iehkdbc m]he ijb \_jrbg_jZ\_gθZklhjhgZhkgh\ZgbyjZ\gZZGZc^bl_ Z m]eu khklZ\ey_fu_ k hkgh\Zgb_f [hdh\uf j_[jhf b [hdh\hc ]jZgvx Z lZd`_ ^\m]jZgguc m]he f_`^m khk_^gbfb [hdh\ufb ]jZgyfb (10 deZkk [ \ukhlmibjZfb^ubiehsZ^v_zih\_joghklbdeZkk \ h[tzfibjZfb^udeZkk ] h[tzf rZjZ hibkZggh]h hdheh ibjZfb^u b iehsZ^v _]h kn_ju (11 deZkk ^ h[tzf rZjZ \ibkZggh]h \ ibjZfb^m b iehsZ^v _]h kn_ju deZkk _ h[tzf ihem\ibkZggh]h \ ibjZfb^m rZjZ b iehsZ^v _]h kn_ju (11 deZkk AZ^ZqZ Hlghr_gb_ iheghc ih\_joghklb ijZ\bevghc n-m]hevghc ibjZfb^udiehsZ^bhkgh\ZgbyjZ\ghlGZc^bl_ Z m]eu khklZ\ey_fu_ k hkgh\Zgb_f [hdh\uf j_[jhf b [hdh\hc ]jZgvx iehkdbc m]he ijb \_jrbg_ Z lZd`_ ^\m]jZgguc m]he f_`^m khk_^gbfb[hdh\ufb]jZgyfbdeZkk [ hlghr_gb_h[tzfh\ibjZfb^ub\ibkZggh]h\g_zrZjZdeZkk \ hlghr_gb_ h[tzfh\ ibjZfb^u b hibkZggh]h hdheh g_z rZjZ (11 deZkk ] hlghr_gb_ h[tzfh\ ibjZfb^u b ihem\ibkZggh]h \ g_z rZjZ (11 deZkk

50

EBL?J:LMJ: 1. :df_heh]by/Ih^j_^G<Dmavfbghc:FAbfbq_\ZKI[Ba^-\h

K-I_l_j[mj]:df_heh]:dZ^k 2. :e_dkZg^jh\ : > <_jg_j : E Ju`bd < B =_hf_ljby ^ey —11 deZkkh\ Mq_[ ihkh[b_ ^ey mqZsboky rd b deZkkh\ k m]em[e bamq fZl_fZlbdb—FIjhk\_s_gb_]— k 3. Zx mjhdb fZl_fZlbdb Dg ^ey mqbl_ey ba hiulZ jZ[hlu—FIjhk\_s_gb_.— k 10. Djml_pdbc < : Ikboheh]by fZl_fZlbq_kdbo kihkh[ghkl_c rdhevgbdh\—FIjhk\_s_gb_68. — k 11. DmavfbgZ G < Ijhn_kkbhgZebaf ^_yl_evghklb ij_ih^Z\Zl_ey— F hjhn__\ b ^j Khkl < B Fbrbg—FIjhk\_s_gb_—k 16. F_lh^u kbkl_fgh]h i_^Z]h]bq_kdh]h bkke_^h\Zgby Ih^ j_^ G < Dmavfbghc— EBa^-\hE=M— k

51

17. IZih\kdbc < F M]em[e_ggh_ bamq_gb_ ]_hf_ljbb \ — deZkkZo

F_lh^S_dhf_g^Zpbbdij_ih^Z\ZgbxdmjkZ]_hf_ljbb\ — deih mq_[ ihkh[bx : > :e_dkZg^jh\Z : E <_jg_jZ < B Ju`bdZ— F Ijhk\_s_gb_— k 18. IhcZ>DZdj_rZlvaZ^Zqb—FGZmdZ. — k 19. Ju`bd<Bmjhdh\fZl_fZlbdb—FIjhk\_s_gb_— k 20. K[hjgbd aZ^Zq ih fZl_fZlbd_ ^ey ihklmiZxsbo \h \lmau Mq_[ ihkh[b_Ih^j_^FBKdZgZ\b—F
ChklZ\bl_eb>hgph\ AZdZa hl ] LbjZ` wda EZ[hjZlhjby hi_jZlb\ghc iheb]jZnbb<=M 52

Проектирование уроков развивающего обучения по геометрии в старших классах: Методические указания

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