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Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 1
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 1
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɡɚɞɚɧɢɣ ɫ ɪɚɡɜɺɪɧɭɬɵɦ ɨɬɜɟɬɨɦ
Ɇɨɞɭɥɶ "Ⱥɥɝɟɛɪɚ"
21
ɍɩɪɨɫɬɢɬɟ ɜɵɪɚɠɟɧɢɟ
10 2 ˜
24
10 2
.
Ɋɟɲɟɧɢɟ.
( 10 2)( 10 2)
1
1
10 2 ˜
10 2
10 4
6
.
2
24
24
24
24
4
1
Ɉɬɜɟɬ: .
2
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
2
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
1
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
2
Ʉɨɦɦɟɧɬɚɪɢɣ. Ɉɲɢɛɤɢ ɜ ɩɪɢɦɟɧɟɧɢɢ ɮɨɪɦɭɥ ɫɱɢɬɚɸɬɫɹ ɫɭɳɟɫɬɜɟɧɧɵɦɢ; ɩɪɢ ɢɯ
ɧɚɥɢɱɢɢ ɪɟɲɟɧɢɟ ɧɟ ɡɚɫɱɢɬɵɜɚɟɬɫɹ.
Ɉɞɢɧ ɢɡ ɤɨɪɧɟɣ ɭɪɚɜɧɟɧɢɹ 3x 2 5x 2m 0 ɪɚɜɟɧ 1. ɇɚɣɞɢɬɟ ɜɬɨɪɨɣ ɤɨɪɟɧɶ.
Ɋɟɲɟɧɢɟ.
ɉɨɞɫɬɚɜɢɦ ɢɡɜɟɫɬɧɵɣ ɤɨɪɟɧɶ ɜ ɭɪɚɜɧɟɧɢɟ: 3 5 2m 0. ɉɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ
ɨɬɧɨɫɢɬɟɥɶɧɨ m.Ɋɟɲɢɦ ɟɝɨ: 2m 2; m 1. ɉɨɞɫɬɚɜɢɦ m ɜ ɭɪɚɜɧɟɧɢɟ: 3x 2 5x 2 0,
ɨɬɤɭɞɚ
5 r 25 4 ˜ 3 ˜ 2
5 r 1
2
, x1 1, x2 .
x
6
6
3
2
Ɉɬɜɟɬ: .
3
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
3
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
2
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
3
22
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 1
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 2
ɇɚɣɞɢɬɟ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ ɜɵɪɚɠɟɧɢɹ ɢ ɡɧɚɱɟɧɢɹ x ɢ y, ɩɪɢ ɤɨɬɨɪɵɯ ɨɧɨ
ɞɨɫɬɢɝɚɟɬɫɹ 6x 5y 7 2x 3y 1 .
Ɋɟɲɟɧɢɟ.
ɋɭɦɦɚ 6x 5y 7 2x 3y 1 ɩɪɢɧɢɦɚɟɬ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ, ɪɚɜɧɨɟ 0, ɬɨɥɶɤɨ ɜ
ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɨɛɚ ɫɥɚɝɚɟɦɵɯ ɨɞɧɨɜɪɟɦɟɧɧɨ ɪɚɜɧɵ 0. ɉɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ
­ 6x 5y 7 0,
® 2x 3y 1 0.
¯
Ɋɟɲɢɦ ɟɺ:
­ 6x 5y 7 0, ­ 4y 4 0,
­ y 1,
­ y 1,
® 6x 9y 3 0; ® 6x 9y 3 0; ®
®
6x
12
0;
¯ x 2.
¯
¯
¯
Ɉɬɜɟɬ: 0; (–2;1).
23
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
4
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
3
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
4
Ɇɨɞɭɥɶ "Ƚɟɨɦɟɬɪɢɹ"
24
ɇɚɣɞɢɬɟ ɜɟɥɢɱɢɧɭ ɭɝɥɚ AOE, ɟɫɥɢ OE – ɛɢɫɫɟɤɬɪɢɫɚ ɭɝɥɚ AOC, OD –
ɛɢɫɫɟɤɬɪɢɫɚ ɭɝɥɚ COB.
Ɋɟɲɟɧɢɟ.
ӘCOB=2·25°=50°; ӘAOC=180°–50°=130°; ӘAOE=130°:2=65°.
Ɉɬɜɟɬ: 65°.
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 1
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 3
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
Ɋɟɲɟɧɢɟ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
2
Ⱦɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɟɬɨɦ
1
ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
2
ȼ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɟ ABCD ɬɨɱɤɚ M — ɫɟɪɟɞɢɧɚ ɫɬɨɪɨɧɵ AB. ɂɡɜɟɫɬɧɨ, ɱɬɨ
MC MD. Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ɞɚɧɧɵɣ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦ — ɩɪɹɦɨɭɝɨɥɶɧɢɤ.
ɉɭɫɬɶ ɬɨɱɤɚ M – ɫɟɪɟɞɢɧɚ ɫɬɨɪɨɧɵ AB ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ABCD – ɪɚɜɧɨɭɞɚɥɟɧɚ ɨɬ ɟɝɨ
ɜɟɪɲɢɧ C ɢ D. Ɍɨɝɞɚ, ɬɪɟɭɝɨɥɶɧɢɤ CMD – ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ, ɩɨɷɬɨɦɭ ‘ MCD ‘ MDC .
AB,
ɉɨɫɤɨɥɶɤɭ
ɩɪɹɦɚɹ CD ɩɚɪɚɥɥɟɥɶɧɚ
ɫɬɨɪɨɧɟ
ɬɨ ‘ BMC ‘ MCD ɢ
‘ AMD ‘ MDC ɤɚɤ ɧɚɤɪɟɫɬ ɥɟɠɚɳɢɟ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, +BMC +AMD ɩɨ ɩɟɪɜɨɦɭ
ɩɪɢɡɧɚɤɭ ɪɚɜɟɧɫɬɜɚ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ( ‘ BMC ‘ AMD, AM BM , MC MD).
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 1
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 4
Ɉɫɧɨɜɚɧɢɟ AC ɪɚɜɧɨɛɟɞɪɟɧɧɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ABC ɪɚɜɧɨ 10. Ɉɤɪɭɠɧɨɫɬɶ
ɪɚɞɢɭɫɚ 7, 5 ɫ ɰɟɧɬɪɨɦ ɜɧɟ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ɤɚɫɚɟɬɫɹ ɩɪɨɞɨɥɠɟɧɢɹ ɛɨɤɨɜɵɯ
ɫɬɨɪɨɧ ɬɪɟɭɝɨɥɶɧɢɤɚ ɢ ɤɚɫɚɟɬɫɹ ɨɫɧɨɜɚɧɢɹ AC ɜ ɟɝɨ ɫɟɪɟɞɢɧɟ. ɇɚɣɞɢɬɟ ɪɚɞɢɭɫ
ɨɤɪɭɠɧɨɫɬɢ, ɜɩɢɫɚɧɧɨɣ ɜ ɬɪɟɭɝɨɥɶɧɢɤ ABC.
Ɋɟɲɟɧɢɟ.
26
25
Ɂɧɚɱɢɬ, ‘ CBM ‘ DAM . ɂɯ ɫɭɦɦɚ ɪɚɜɧɚ 180q, ɬ.ɤ. ɷɬɨ ɞɜɚ ɭɝɥɚ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ,
ɩɪɢɥɟɠɚɳɢɟ ɤ ɨɞɧɨɣ ɫɬɨɪɨɧɟ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ‘ CBM ‘ DAM 90D. ɉɨ ɫɜɨɣɫɬɜɭ
ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ɭɝɥɵ BCD ɢ CDA ɬɚɤɠɟ ɩɪɹɦɵɟ. Ɂɧɚɱɢɬ, ABCD – ɩɪɹɦɨɭɝɨɥɶɧɢɤ.
Ʉɨɦɦɟɧɬɚɪɢɣ: Ɋɚɜɟɧɫɬɜɨ ɬɪɟɭɝɨɥɶɧɢɤɨɜ BMC ɢ AMD ɦɨɠɟɬ ɛɵɬɶ ɞɨɤɚɡɚɧɨ ɢɧɚɱɟ,
ɧɚɩɪɢɦɟɪ, ɩɨ ɬɪɟɬɶɟɦɭ ɩɪɢɡɧɚɤɭ ɪɚɜɟɧɫɬɜɚ ɬɪɟɭɝɨɥɶɧɢɤɨɜ.
Ⱦɪɭɝɨɟ ɜɨɡɦɨɠɧɨɟ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ:
ɉɭɫɬɶ ɬɨɱɤɚ O – ɫɟɪɟɞɢɧɚ CD. ɑɟɬɵɪɟɯɭɝɨɥɶɧɢɤ OMBC ɹɜɥɹɟɬɫɹ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɨɦ,
ɩɨɫɤɨɥɶɤɭ ɟɝɨ ɫɬɨɪɨɧɵ OC ɢ MB ɩɚɪɚɥɥɟɥɶɧɵ ɢ ɪɚɜɧɵ. Ɍɪɟɭɝɨɥɶɧɢɤ MCD –
ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ, ɩɨɷɬɨɦɭ OM – ɟɝɨ ɜɵɫɨɬɚ. Ɂɧɚɱɢɬ, OMBC – ɩɪɹɦɨɭɝɨɥɶɧɢɤ,
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɝɨɥ CBM – ɩɪɹɦɨɣ.
Ⱦɚɧɧɚɹ ɨɤɪɭɠɧɨɫɬɶ ɤɚɫɚɟɬɫɹ ɫɬɨɪɨɧɵ AC ɜ ɟɺ ɫɟɪɟɞɢɧɟ M ɢ ɩɪɨɞɨɥɠɟɧɢɣ ɫɬɨɪɨɧ BA ɢ
BC ɬɪɟɭɝɨɥɶɧɢɤɚ ABC.
ɉɭɫɬɶ O — ɰɟɧɬɪ ɷɬɨɣ ɨɤɪɭɠɧɨɫɬɢ, ɚ Q — ɰɟɧɬɪ ɨɤɪɭɠɧɨɫɬɢ, ɜɩɢɫɚɧɧɨɣ ɜ
ɬɪɟɭɝɨɥɶɧɢɤ ABC. ɍɝɨɥ OAQ – ɩɪɹɦɨɣ ɤɚɤ ɭɝɨɥ ɦɟɠɞɭ ɛɢɫɫɟɤɬɪɢɫɚɦɢ ɫɦɟɠɧɵɯ ɭɝɥɨɜ.
Ɍɪɟɭɝɨɥɶɧɢɤ OAQ – ɩɪɹɦɨɭɝɨɥɶɧɵɣ, AM – ɟɝɨ ɜɵɫɨɬɚ. ɂɡ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ɧɚɯɨɞɢɦ,
AM 2 10
ɱɬɨ AM 2 MQ ˜ MO. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, QM
.
3
OM
10
.
Ɉɬɜɟɬ:
3
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ
4
ɨɬɜɟɬ
ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɧɨ ɞɚɧɵ
3
ɧɟɩɨɥɧɵɟ ɨɛɴɹɫɧɟɧɢɹ ɢɥɢ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɨɲɢɛɤɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
4
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɜɟɪɧɨɟ, ɜɫɟ ɲɚɝɢ ɨɛɨɫɧɨɜɚɧɵ
3
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɫɨɞɟɪɠɢɬ ɧɟɬɨɱɧɨɫɬɢ ɢɥɢ ɩɪɨɛɟɥɵ, ɧɚɩɪɢɦɟɪ, ɨɬɫɭɬɫɬɜɭɸɬ
2
ɫɫɵɥɤɢ ɧɚ ɫɜɨɣɫɬɜɚ ɩɚɪɚɥɥɟɥɶɧɵɯ ɩɪɹɦɵɯ ɢɥɢ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
3
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 2
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 1
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɡɚɞɚɧɢɣ ɫ ɪɚɡɜɺɪɧɭɬɵɦ ɨɬɜɟɬɨɦ
ɍɩɪɨɫɬɢɬɟ ɜɵɪɚɠɟɧɢɟ
54
15 3 ˜
15 3
.
Ɋɟɲɟɧɢɟ.
54
15 3 ˜
Ɉɬɜɟɬ: 3.
54
15 3
( 15 3)( 15 3)
54
15 9
54
6
9
3.
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
2
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
1
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
2
Ʉɨɦɦɟɧɬɚɪɢɣ. Ɉɲɢɛɤɢ ɜ ɩɪɢɦɟɧɟɧɢɢ ɮɨɪɦɭɥ ɫɱɢɬɚɸɬɫɹ ɫɭɳɟɫɬɜɟɧɧɵɦɢ; ɩɪɢ ɢɯ
ɧɚɥɢɱɢɢ ɪɟɲɟɧɢɟ ɧɟ ɡɚɫɱɢɬɵɜɚɟɬɫɹ.
Ɉɞɢɧ ɢɡ ɤɨɪɧɟɣ ɭɪɚɜɧɟɧɢɹ 5x 2 2x 3p
Ɋɟɲɟɧɢɟ.
ɉɨɞɫɬɚɜɢɦ ɢɡɜɟɫɬɧɵɣ ɤɨɪɟɧɶ ɜ ɭɪɚɜɧɟɧɢɟ:
ɨɬɧɨɫɢɬɟɥɶɧɨ p. Ɋɟɲɢɦ ɟɝɨ: 3p 3; p
5x 2 2x 3 0, ɨɬɤɭɞɚ
2r 44˜5˜3
2r8
,
x
10
10
22
0 ɪɚɜɟɧ 1. ɇɚɣɞɢɬɟ ɜɬɨɪɨɣ ɤɨɪɟɧɶ.
5 2 3p 0. ɉɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ
1. ɉɨɞɫɬɚɜɢɦ p ɜ ɭɪɚɜɧɟɧɢɟ:
x1
1, x2
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 2
ɇɚɣɞɢɬɟ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ ɜɵɪɚɠɟɧɢɹ ɢ ɡɧɚɱɟɧɢɹ x ɢ y, ɩɪɢ ɤɨɬɨɪɵɯ ɨɧɨ
ɞɨɫɬɢɝɚɟɬɫɹ: 3x 4y 2 x 5y 3 .
Ɋɟɲɟɧɢɟ.
ɋɭɦɦɚ 3x 4y 2 x 5y 3 ɩɪɢɧɢɦɚɟɬ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ, ɪɚɜɧɨɟ 0, ɬɨɥɶɤɨ ɜ
ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɨɛɚ ɫɥɚɝɚɟɦɵɯ ɨɞɧɨɜɪɟɦɟɧɧɨ ɪɚɜɧɵ 0. ɉɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ
­ 3x 4y 2 0,
® x 5y 3 0.
¯
Ɋɟɲɢɦ ɟɺ:
­ 3x 4y 2 0, ­ 11y 11 0, ­ y 1,
® 3x 15y 9 0, ® x 5y 3 0, ®
¯ x 2.
¯
¯
Ɉɬɜɟɬ: 0; (2;1).
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
4
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
3
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
4
23
Ɇɨɞɭɥɶ "Ⱥɥɝɟɛɪɚ"
21
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 2
0, 6.
Ɉɬɜɟɬ: 0, 6.
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
3
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
2
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
3
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɨɞɭɥɶ "Ƚɟɨɦɟɬɪɢɹ"
24
ɇɚɣɞɢɬɟ ɜɟɥɢɱɢɧɭ ɭɝɥɚ COE, ɟɫɥɢ OE – ɛɢɫɫɟɤɬɪɢɫɚ ɭɝɥɚ AOC, OD –
ɛɢɫɫɟɤɬɪɢɫɚ ɭɝɥɚ COB.
Ɋɟɲɟɧɢɟ.
ӘCOB=2·35°=70°; ӘAOC=180°–70°=110°; ӘɋOE=110°:2=55°.
Ɉɬɜɟɬ: 55°.
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 2
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 3
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
Ɋɟɲɟɧɢɟ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
2
Ⱦɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɟɬɨɦ
1
ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
2
ȼ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɟ ABCD ɬɨɱɤɚ K — ɫɟɪɟɞɢɧɚ ɫɬɨɪɨɧɵ AB. ɂɡɜɟɫɬɧɨ, ɱɬɨ
KC KD. Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ɞɚɧɧɵɣ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦ — ɩɪɹɦɨɭɝɨɥɶɧɢɤ.
ɉɭɫɬɶ ɬɨɱɤɚ K – ɫɟɪɟɞɢɧɚ ɫɬɨɪɨɧɵ AB ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ABCD – ɪɚɜɧɨɭɞɚɥɟɧɚ ɨɬ ɟɝɨ
ɜɟɪɲɢɧ C ɢ D. Ɍɨɝɞɚ, ɬɪɟɭɝɨɥɶɧɢɤ CKD – ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ, ɩɨɷɬɨɦɭ ‘ KCD ‘ KDC .
ɉɨɫɤɨɥɶɤɭ ɩɪɹɦɚɹ CD ɩɚɪɚɥɥɟɥɶɧɚ ɫɬɨɪɨɧɟ AB, ɬɨ ‘ BKC ‘ KCD ɢ ‘ AKD ‘ KDC
ɤɚɤ ɧɚɤɪɟɫɬ ɥɟɠɚɳɢɟ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, +BKC +AKD ɩɨ ɩɟɪɜɨɦɭ ɩɪɢɡɧɚɤɭ ɪɚɜɟɧɫɬɜɚ
ɬɪɟɭɝɨɥɶɧɢɤɨɜ ( ‘ BKC ‘ AKD, AK BK , KC KD).
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 2
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 4
Ɉɫɧɨɜɚɧɢɟ AC ɪɚɜɧɨɛɟɞɪɟɧɧɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ABC ɪɚɜɧɨ 12. Ɉɤɪɭɠɧɨɫɬɶ
ɪɚɞɢɭɫɚ 8 ɫ ɰɟɧɬɪɨɦ ɜɧɟ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ɤɚɫɚɟɬɫɹ ɩɪɨɞɨɥɠɟɧɢɹ ɛɨɤɨɜɵɯ
ɫɬɨɪɨɧ ɬɪɟɭɝɨɥɶɧɢɤɚ ɢ ɤɚɫɚɟɬɫɹ ɨɫɧɨɜɚɧɢɹ AC ɜ ɟɝɨ ɫɟɪɟɞɢɧɟ. ɇɚɣɞɢɬɟ ɪɚɞɢɭɫ
ɨɤɪɭɠɧɨɫɬɢ, ɜɩɢɫɚɧɧɨɣ ɜ ɬɪɟɭɝɨɥɶɧɢɤ ABC.
Ɋɟɲɟɧɢɟ.
26
25
Ɂɧɚɱɢɬ, ‘ CBK ‘ DAK . ɂɯ ɫɭɦɦɚ ɪɚɜɧɚ 180q, ɬ.ɤ. ɷɬɨ ɞɜɚ ɭɝɥɚ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ,
ɩɪɢɥɟɠɚɳɢɟ ɤ ɨɞɧɨɣ ɫɬɨɪɨɧɟ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ‘ CBK ‘ DAK 90D. ɉɨ ɫɜɨɣɫɬɜɭ
ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ɭɝɥɵ BCD ɢ CDA ɬɚɤɠɟ ɩɪɹɦɵɟ. Ɂɧɚɱɢɬ, ABCD – ɩɪɹɦɨɭɝɨɥɶɧɢɤ.
Ʉɨɦɦɟɧɬɚɪɢɣ: Ɋɚɜɟɧɫɬɜɨ ɬɪɟɭɝɨɥɶɧɢɤɨɜ BKC ɢ AKD ɦɨɠɟɬ ɛɵɬɶ ɞɨɤɚɡɚɧɨ ɢɧɚɱɟ,
ɧɚɩɪɢɦɟɪ, ɩɨ ɬɪɟɬɶɟɦɭ ɩɪɢɡɧɚɤɭ ɪɚɜɟɧɫɬɜɚ ɬɪɟɭɝɨɥɶɧɢɤɨɜ.
Ⱦɪɭɝɨɟ ɜɨɡɦɨɠɧɨɟ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ:
ɉɭɫɬɶ ɬɨɱɤɚ O – ɫɟɪɟɞɢɧɚ CD. ɑɟɬɵɪɟɯɭɝɨɥɶɧɢɤ OKBC ɹɜɥɹɟɬɫɹ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɨɦ,
ɩɨɫɤɨɥɶɤɭ ɟɝɨ ɫɬɨɪɨɧɵ OC ɢ KB ɩɚɪɚɥɥɟɥɶɧɵ ɢ ɪɚɜɧɵ. Ɍɪɟɭɝɨɥɶɧɢɤ KCD –
ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ, ɩɨɷɬɨɦɭ OK – ɟɝɨ ɜɵɫɨɬɚ. Ɂɧɚɱɢɬ, OKBC – ɩɪɹɦɨɭɝɨɥɶɧɢɤ,
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɝɨɥ CBK – ɩɪɹɦɨɣ.
Ⱦɚɧɧɚɹ ɨɤɪɭɠɧɨɫɬɶ ɤɚɫɚɟɬɫɹ ɫɬɨɪɨɧɵ AC ɜ ɟɺ ɫɟɪɟɞɢɧɟ M ɢ ɩɪɨɞɨɥɠɟɧɢɣ ɫɬɨɪɨɧ BA ɢ
BC ɬɪɟɭɝɨɥɶɧɢɤɚ ABC.
ɉɭɫɬɶ O — ɰɟɧɬɪ ɷɬɨɣ ɨɤɪɭɠɧɨɫɬɢ, ɚ Q — ɰɟɧɬɪ ɨɤɪɭɠɧɨɫɬɢ, ɜɩɢɫɚɧɧɨɣ ɜ
ɬɪɟɭɝɨɥɶɧɢɤ ABC. ɍɝɨɥ OAQ – ɩɪɹɦɨɣ ɤɚɤ ɭɝɨɥ ɦɟɠɞɭ ɛɢɫɫɟɤɬɪɢɫɚɦɢ ɫɦɟɠɧɵɯ ɭɝɥɨɜ.
Ɍɪɟɭɝɨɥɶɧɢɤ OAQ – ɩɪɹɦɨɭɝɨɥɶɧɵɣ, AM – ɟɝɨ ɜɵɫɨɬɚ. ɂɡ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ɧɚɯɨɞɢɦ,
AM 2 9
ɱɬɨ AM 2 MQ ˜ MO. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, QM
4, 5.
2
OM
Ɉɬɜɟɬ: 4, 5.
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ
4
ɨɬɜɟɬ
ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɧɨ ɞɚɧɵ
3
ɧɟɩɨɥɧɵɟ ɨɛɴɹɫɧɟɧɢɹ ɢɥɢ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɨɲɢɛɤɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
4
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɜɟɪɧɨɟ, ɜɫɟ ɲɚɝɢ ɨɛɨɫɧɨɜɚɧɵ
3
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɫɨɞɟɪɠɢɬ ɧɟɬɨɱɧɨɫɬɢ ɢɥɢ ɩɪɨɛɟɥɵ, ɧɚɩɪɢɦɟɪ, ɨɬɫɭɬɫɬɜɭɸɬ
2
ɫɫɵɥɤɢ ɧɚ ɫɜɨɣɫɬɜɚ ɩɚɪɚɥɥɟɥɶɧɵɯ ɩɪɹɦɵɯ ɢɥɢ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
3
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 3
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 1
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɡɚɞɚɧɢɣ ɫ ɪɚɡɜɺɪɧɭɬɵɦ ɨɬɜɟɬɨɦ
Ɇɨɞɭɥɶ "Ⱥɥɝɟɛɪɚ"
21
ɍɩɪɨɫɬɢɬɟ ɜɵɪɚɠɟɧɢɟ
15 3 ˜
24
15 3
.
Ɋɟɲɟɧɢɟ.
( 15 3)( 15 3)
1
1
15 3 ˜
15 3
15 9
6
.
24
24
24
24
4
2
1
Ɉɬɜɟɬ: .
2
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
2
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
1
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
2
Ʉɨɦɦɟɧɬɚɪɢɣ. Ɉɲɢɛɤɢ ɜ ɩɪɢɦɟɧɟɧɢɢ ɮɨɪɦɭɥ ɫɱɢɬɚɸɬɫɹ ɫɭɳɟɫɬɜɟɧɧɵɦɢ; ɩɪɢ ɢɯ
ɧɚɥɢɱɢɢ ɪɟɲɟɧɢɟ ɧɟ ɡɚɫɱɢɬɵɜɚɟɬɫɹ.
Ɉɞɢɧ ɢɡ ɤɨɪɧɟɣ ɭɪɚɜɧɟɧɢɹ 4x x 3m 0 ɪɚɜɟɧ 1. ɇɚɣɞɢɬɟ ɜɬɨɪɨɣ ɤɨɪɟɧɶ.
Ɋɟɲɟɧɢɟ.
ɉɨɞɫɬɚɜɢɦ ɢɡɜɟɫɬɧɵɣ ɤɨɪɟɧɶ ɜ ɭɪɚɜɧɟɧɢɟ: 4 1 3m 0. ɉɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ
ɨɬɧɨɫɢɬɟɥɶɧɨ m. Ɋɟɲɢɦ ɟɝɨ: 3m 3; m 1. ɉɨɞɫɬɚɜɢɦ m ɜ ɭɪɚɜɧɟɧɢɟ:
4x 2 x 3 0, ɨɬɤɭɞɚ
1r7
1r 14˜3˜4
3
x
, x1 1, x2 .
8
4
8
3
Ɉɬɜɟɬ: .
4
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
3
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
2
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
3
22
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 3
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 2
ɇɚɣɞɢɬɟ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ ɜɵɪɚɠɟɧɢɹ ɢ ɡɧɚɱɟɧɢɹ x ɢ y, ɩɪɢ ɤɨɬɨɪɵɯ ɨɧɨ
ɞɨɫɬɢɝɚɟɬɫɹ 6x y 5 3x 2y 1 .
Ɋɟɲɟɧɢɟ.
ɋɭɦɦɚ 6x y 5 3x 2y 1 ɩɪɢɧɢɦɚɟɬ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ, ɪɚɜɧɨɟ 0, ɬɨɥɶɤɨ ɜ
ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɨɛɚ ɫɥɚɝɚɟɦɵɯ ɨɞɧɨɜɪɟɦɟɧɧɨ ɪɚɜɧɵ 0. ɉɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ
­ 6x y 5 0,
® 3x 2y 1 0.
¯
Ɋɟɲɢɦ ɟɺ
­ 6x y 5 0, ­ 3y 3 0,
­ y 1,
­ y 1,
® 6x 4y 2 0, ® 6x y 5 0, ®
®
6x
6
0,
¯ x 1.
¯
¯
¯
Ɉɬɜɟɬ: 0; (–1;1).
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
4
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
3
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
4
23
Ɇɨɞɭɥɶ "Ƚɟɨɦɟɬɪɢɹ"
2
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
24
ɇɚɣɞɢɬɟ ɜɟɥɢɱɢɧɭ ɭɝɥɚ COE, ɟɫɥɢ OE – ɛɢɫɫɟɤɬɪɢɫɚ ɭɝɥɚ AOC, OD –
ɛɢɫɫɟɤɬɪɢɫɚ ɭɝɥɚ COB.
Ɋɟɲɟɧɢɟ.
ӘCOB=2·32°=64°; ӘAOC=180°–64°=116°; ӘɋOE=116°:2=58°.
Ɉɬɜɟɬ: 58°.
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
Ɋɟɲɟɧɢɟ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
2
Ⱦɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɟɬɨɦ
1
ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
2
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 3
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 3
ɉɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɭɝɥɵ ɱɟɬɵɪɟɯɭɝɨɥɶɧɢɤɚ ɩɨɩɚɪɧɨ ɪɚɜɧɵ. Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ɷɬɨɬ
ɱɟɬɵɪɟɯɭɝɨɥɶɧɢɤ – ɩɚɪɚɥɥɟɥɨɝɪɚɦɦ.
ɉɭɫɬɶ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɭɝɥɵ A ɢ C ɱɟɬɵɪɟɯɭɝɨɥɶɧɢɤɚ ABCD ɪɚɜɧɵ Į , ɚ
ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɭɝɥɵ B ɢ D ɪɚɜɧɵ ȕ. ɉɨɫɤɨɥɶɤɭ ɫɭɦɦɚ ɭɝɥɨɜ ɥɸɛɨɝɨ
ɱɟɬɵɪɟɯɭɝɨɥɶɧɢɤɚ ɪɚɜɧɚ 360q, ɬɨ 2Į 2ȕ 360q. Ɂɧɚɱɢɬ, Į ȕ 180q. Ɍɚɤ ɤɚɤ ɫɭɦɦɚ
ɜɧɭɬɪɟɧɧɢɯ ɨɞɧɨɫɬɨɪɨɧɧɢɯ ɭɝɥɨɜ ɩɪɢ ɫɟɤɭɳɟɣ ɪɚɜɧɚ 180D,ɬɨ ɩɨ ɩɪɢɡɧɚɤɭ
ɩɚɪɚɥɥɟɥɶɧɵɯ ɩɪɹɦɵɯ AB ɩɚɪɚɥɥɟɥɶɧɚ CD, BC ɩɚɪɚɥɥɟɥɶɧɚ AD. Ɂɧɚɱɢɬ,
ɱɟɬɵɪɺɯɭɝɨɥɶɧɢɤ ABCD ɹɜɥɹɟɬɫɹ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɨɦ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ.
25
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 3
Ɉɬɜɟɬ:
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 4
8
.
3
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ
4
ɨɬɜɟɬ
ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɧɨ ɞɚɧɵ
3
ɧɟɩɨɥɧɵɟ ɨɛɴɹɫɧɟɧɢɹ ɢɥɢ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɨɲɢɛɤɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
4
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɜɟɪɧɨɟ, ɜɫɟ ɲɚɝɢ ɨɛɨɫɧɨɜɚɧɵ
3
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɫɨɞɟɪɠɢɬ ɧɟɬɨɱɧɨɫɬɢ ɢɥɢ ɩɪɨɛɟɥɵ, ɧɚɩɪɢɦɟɪ, ɨɬɫɭɬɫɬɜɭɸɬ
2
ɫɫɵɥɤɢ ɧɚ ɫɜɨɣɫɬɜɚ ɩɚɪɚɥɥɟɥɶɧɵɯ ɩɪɹɦɵɯ ɢɥɢ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
3
Ɉɫɧɨɜɚɧɢɟ AC ɪɚɜɧɨɛɟɞɪɟɧɧɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ABC ɪɚɜɧɨ 8. Ɉɤɪɭɠɧɨɫɬɶ
ɪɚɞɢɭɫɚ 6 ɫ ɰɟɧɬɪɨɦ ɜɧɟ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ɤɚɫɚɟɬɫɹ ɩɪɨɞɨɥɠɟɧɢɹ ɛɨɤɨɜɵɯ
ɫɬɨɪɨɧ ɬɪɟɭɝɨɥɶɧɢɤɚ ɢ ɤɚɫɚɟɬɫɹ ɨɫɧɨɜɚɧɢɹ AC ɜ ɟɝɨ ɫɟɪɟɞɢɧɟ. ɇɚɣɞɢɬɟ ɪɚɞɢɭɫ
ɨɤɪɭɠɧɨɫɬɢ, ɜɩɢɫɚɧɧɨɣ ɜ ɬɪɟɭɝɨɥɶɧɢɤ ABC.
Ɋɟɲɟɧɢɟ.
26
Ⱦɚɧɧɚɹ ɨɤɪɭɠɧɨɫɬɶ ɤɚɫɚɟɬɫɹ ɫɬɨɪɨɧɵ AC ɜ ɟɺ ɫɟɪɟɞɢɧɟ M ɢ ɩɪɨɞɨɥɠɟɧɢɣ ɫɬɨɪɨɧ BA ɢ
BC ɬɪɟɭɝɨɥɶɧɢɤɚ ABC.
ɉɭɫɬɶ O – ɰɟɧɬɪ ɷɬɨɣ ɨɤɪɭɠɧɨɫɬɢ, ɚ Q – ɰɟɧɬɪ ɨɤɪɭɠɧɨɫɬɢ, ɜɩɢɫɚɧɧɨɣ ɜ ɬɪɟɭɝɨɥɶɧɢɤ
ABC. ɍɝɨɥ OAQ – ɩɪɹɦɨɣ ɤɚɤ ɭɝɨɥ ɦɟɠɞɭ ɛɢɫɫɟɤɬɪɢɫɚɦɢ ɫɦɟɠɧɵɯ ɭɝɥɨɜ. Ɍɪɟɭɝɨɥɶɧɢɤ
OAQ – ɩɪɹɦɨɭɝɨɥɶɧɵɣ, AM – ɟɝɨ ɜɵɫɨɬɚ. ɂɡ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ɧɚɯɨɞɢɦ, ɱɬɨ
AM 2 8
.
AM 2 MQ ˜ MO. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, QM
3
OM
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 4
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 1
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɡɚɞɚɧɢɣ ɫ ɪɚɡɜɺɪɧɭɬɵɦ ɨɬɜɟɬɨɦ
ɍɩɪɨɫɬɢɬɟ ɜɵɪɚɠɟɧɢɟ
54
31 5 ˜
31 5
.
Ɋɟɲɟɧɢɟ.
54
31 5 ˜
Ɉɬɜɟɬ: 3.
54
31 5
( 31 5)( 31 5)
54
31 25
54
6
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 2
ɇɚɣɞɢɬɟ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ ɜɵɪɚɠɟɧɢɹ ɢ ɡɧɚɱɟɧɢɹ x ɢ y, ɩɪɢ ɤɨɬɨɪɵɯ ɨɧɨ
ɞɨɫɬɢɝɚɟɬɫɹ: 3x 4y 1 x 5y 6 .
Ɋɟɲɟɧɢɟ.
ɋɭɦɦɚ 3x 4y 1 x 5y 6 ɩɪɢɧɢɦɚɟɬ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ, ɪɚɜɧɨɟ 0, ɬɨɥɶɤɨ ɜ
ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɨɛɚ ɫɥɚɝɚɟɦɵɯ ɨɞɧɨɜɪɟɦɟɧɧɨ ɪɚɜɧɵ 0. ɉɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ
­ 3x 4y 1 0,
® x 5y 6 0.
¯
Ɋɟɲɢɦ ɟɺ
­ 3x 4y 1 0,
­ 19y 19 0, ­ y 1,
® 3x 15y 18 0; ® x 5y 6 0; ®
¯ x 1.
¯
¯
Ɉɬɜɟɬ: 0; (–1;1).
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
4
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
3
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
4
23
Ɇɨɞɭɥɶ "Ⱥɥɝɟɛɪɚ"
21
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 4
9
3.
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
2
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
1
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
2
Ʉɨɦɦɟɧɬɚɪɢɣ. Ɉɲɢɛɤɢ ɜ ɩɪɢɦɟɧɟɧɢɢ ɮɨɪɦɭɥ ɫɱɢɬɚɸɬɫɹ ɫɭɳɟɫɬɜɟɧɧɵɦɢ; ɩɪɢ ɢɯ
ɧɚɥɢɱɢɢ ɪɟɲɟɧɢɟ ɧɟ ɡɚɫɱɢɬɵɜɚɟɬɫɹ.
Ɉɞɢɧ ɢɡ ɤɨɪɧɟɣ ɭɪɚɜɧɟɧɢɹ 5x 2 7x 2m 0 ɪɚɜɟɧ –1. ɇɚɣɞɢɬɟ ɜɬɨɪɨɣ ɤɨɪɟɧɶ.
Ɋɟɲɟɧɢɟ.
ɉɨɞɫɬɚɜɢɦ ɢɡɜɟɫɬɧɵɣ ɤɨɪɟɧɶ ɜ ɭɪɚɜɧɟɧɢɟ: 5 7 2m 0. ɉɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ
ɨɬɧɨɫɢɬɟɥɶɧɨ m. Ɋɟɲɢɦ ɟɝɨ: 2m 2; m 1. ɉɨɞɫɬɚɜɢɦ m ɜ ɭɪɚɜɧɟɧɢɟ:
5x 2 7x 2 0, ɨɬɤɭɞɚ
7 r 49 4 ˜ 5 ˜ 2
7 r 3
, x1 1, x2 0, 4.
x
10
10
Ɉɬɜɟɬ: 0, 4.
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
3
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
2
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
3
Ɇɨɞɭɥɶ "Ƚɟɨɦɟɬɪɢɹ"
22
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
24
ɇɚɣɞɢɬɟ ɜɟɥɢɱɢɧɭ ɭɝɥɚ DOB, ɟɫɥɢ OE – ɛɢɫɫɟɤɬɪɢɫɚ ɭɝɥɚ AOC, OD –
ɛɢɫɫɟɤɬɪɢɫɚ ɭɝɥɚ COB.
Ɋɟɲɟɧɢɟ.
ӘCOȺ=2·64°=128°; ӘȼOC=180°–128°=52°; ӘDOB=52°:2=26°.
Ɉɬɜɟɬ: 26°.
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
Ɋɟɲɟɧɢɟ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
2
Ⱦɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɟɬɨɦ
1
ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
2
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 4
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 3
ɋɟɪɟɞɢɧɚ ɫɬɨɪɨɧɵ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ɪɚɜɧɨɭɞɚɥɟɧɚ ɨɬ ɤɨɧɰɨɜ ɟɝɨ
ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɣ ɫɬɨɪɨɧɵ. Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ɞɚɧɧɵɣ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦ –
ɩɪɹɦɨɭɝɨɥɶɧɢɤ.
ɉɭɫɬɶ ɬɨɱɤɚ O – ɫɟɪɟɞɢɧɚ ɫɬɨɪɨɧɵ BC ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ABCD – ɪɚɜɧɨɭɞɚɥɟɧɚ ɨɬ ɟɝɨ
ɜɟɪɲɢɧ A ɢ D.Ɍɨɝɞɚ ɬɪɟɭɝɨɥɶɧɢɤ AOD ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ, ɩɨɷɬɨɦɭ ‘ AOD ‘ ODA.
ɉɨɫɤɨɥɶɤɭ ɩɪɹɦɚɹ ȼɋ ɩɚɪɚɥɥɟɥɶɧɚ ɫɬɨɪɨɧɟ AD, ɬɨ ɭɝɥɵ BOA ɢ COD ɪɚɜɧɵ ɭɤɚɡɚɧɧɵɦ
ɭɝɥɚɦ ɤɚɤ ɧɚɤɪɟɫɬ ɥɟɠɚɳɢɟ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, +BOA +COD ɩɨ ɩɟɪɜɨɦɭ ɩɪɢɡɧɚɤɭ
ɪɚɜɟɧɫɬɜɚ ɬɪɟɭɝɨɥɶɧɢɤɨɜ. Ɂɧɚɱɢɬ, ‘ ABO ‘ ODA.ɉɭɫɬɶ ɢɯ ɜɟɥɢɱɢɧɚ ɪɚɜɧɚ Į . ɉɪɹɦɵɟ
AB ɢ CD ɩɚɪɚɥɥɟɥɶɧɵ, ɩɨɷɬɨɦɭ Į Į 180q, ɬ.ɟ. Į 90q. ɉɨ ɫɜɨɣɫɬɜɭ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ
ɭɝɥɵ BAD ɢ CDA ɬɚɤɠɟ ɩɪɹɦɵɟ. Ɂɧɚɱɢɬ, ABCD – ɩɪɹɦɨɭɝɨɥɶɧɢɤ.
25
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 4
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 4
ɉɭɫɬɶ O — ɰɟɧɬɪ ɷɬɨɣ ɨɤɪɭɠɧɨɫɬɢ, ɚ Q — ɰɟɧɬɪ ɨɤɪɭɠɧɨɫɬɢ, ɜɩɢɫɚɧɧɨɣ ɜ
ɬɪɟɭɝɨɥɶɧɢɤ ABC. ɍɝɨɥ OAQ – ɩɪɹɦɨɣ ɤɚɤ ɭɝɨɥ ɦɟɠɞɭ ɛɢɫɫɟɤɬɪɢɫɚɦɢ ɫɦɟɠɧɵɯ ɭɝɥɨɜ.
Ɍɪɟɭɝɨɥɶɧɢɤ OAQ – ɩɪɹɦɨɭɝɨɥɶɧɵɣ, AM – ɟɝɨ ɜɵɫɨɬɚ. ɂɡ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ɧɚɯɨɞɢɦ,
AM 2 9
ɱɬɨ AM 2 MQ ˜ MO. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, QM
1, 8.
5
OM
Ɉɬɜɟɬ: 1, 8.
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ
4
ɨɬɜɟɬ
ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɧɨ ɞɚɧɵ
3
ɧɟɩɨɥɧɵɟ ɨɛɴɹɫɧɟɧɢɹ ɢɥɢ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɨɲɢɛɤɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
4
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɜɟɪɧɨɟ, ɜɫɟ ɲɚɝɢ ɨɛɨɫɧɨɜɚɧɵ
3
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɫɨɞɟɪɠɢɬ ɧɟɬɨɱɧɨɫɬɢ ɢɥɢ ɩɪɨɛɟɥɵ, ɧɚɩɪɢɦɟɪ, ɨɬɫɭɬɫɬɜɭɸɬ
2
ɫɫɵɥɤɢ ɧɚ ɫɜɨɣɫɬɜɚ ɩɚɪɚɥɥɟɥɶɧɵɯ ɩɪɹɦɵɯ ɢɥɢ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
3
Ɉɫɧɨɜɚɧɢɟ AC ɪɚɜɧɨɛɟɞɪɟɧɧɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ABC ɪɚɜɧɨ 6. Ɉɤɪɭɠɧɨɫɬɶ
ɪɚɞɢɭɫɚ 5 ɫ ɰɟɧɬɪɨɦ ɜɧɟ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ɤɚɫɚɟɬɫɹ ɩɪɨɞɨɥɠɟɧɢɹ ɛɨɤɨɜɵɯ
ɫɬɨɪɨɧ ɬɪɟɭɝɨɥɶɧɢɤɚ ɢ ɤɚɫɚɟɬɫɹ ɨɫɧɨɜɚɧɢɹ AC ɜ ɟɝɨ ɫɟɪɟɞɢɧɟ. ɇɚɣɞɢɬɟ ɪɚɞɢɭɫ
ɨɤɪɭɠɧɨɫɬɢ, ɜɩɢɫɚɧɧɨɣ ɜ ɬɪɟɭɝɨɥɶɧɢɤ ABC.
Ɋɟɲɟɧɢɟ.
26
Ⱦɚɧɧɚɹ ɨɤɪɭɠɧɨɫɬɶ ɤɚɫɚɟɬɫɹ ɫɬɨɪɨɧɵ AC ɜ ɟɺ ɫɟɪɟɞɢɧɟ M ɢ ɩɪɨɞɨɥɠɟɧɢɣ ɫɬɨɪɨɧ BA ɢ
BC ɬɪɟɭɝɨɥɶɧɢɤɚ ABC.
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 5
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 1
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɡɚɞɚɧɢɣ ɫ ɪɚɡɜɺɪɧɭɬɵɦ ɨɬɜɟɬɨɦ
Ɇɨɞɭɥɶ "Ⱥɥɝɟɛɪɚ"
21
ɍɩɪɨɫɬɢɬɟ ɜɵɪɚɠɟɧɢɟ
10 2 ˜
24
10 2
.
Ɋɟɲɟɧɢɟ.
( 10 2)( 10 2)
1
1
10 2 ˜
10 2
10 4
6
.
2
24
24
24
24
4
1
Ɉɬɜɟɬ: .
2
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
2
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
1
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
2
Ʉɨɦɦɟɧɬɚɪɢɣ. Ɉɲɢɛɤɢ ɜ ɩɪɢɦɟɧɟɧɢɢ ɮɨɪɦɭɥ ɫɱɢɬɚɸɬɫɹ ɫɭɳɟɫɬɜɟɧɧɵɦɢ; ɩɪɢ ɢɯ
ɧɚɥɢɱɢɢ ɪɟɲɟɧɢɟ ɧɟ ɡɚɫɱɢɬɵɜɚɟɬɫɹ.
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 5
ɇɚɣɞɢɬɟ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ ɜɵɪɚɠɟɧɢɹ ɢ ɡɧɚɱɟɧɢɹ x ɢ y, ɩɪɢ ɤɨɬɨɪɵɯ ɨɧɨ
ɞɨɫɬɢɝɚɟɬɫɹ 6x y 5 3x 2y 1 .
Ɋɟɲɟɧɢɟ.
ɋɭɦɦɚ 6x y 5 3x 2y 1 ɩɪɢɧɢɦɚɟɬ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ, ɪɚɜɧɨɟ 0, ɬɨɥɶɤɨ ɜ
ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɨɛɚ ɫɥɚɝɚɟɦɵɯ ɨɞɧɨɜɪɟɦɟɧɧɨ ɪɚɜɧɵ 0. ɉɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ
­ 6x y 5 0,
® 3x 2y 1 0.
¯
Ɋɟɲɢɦ ɟɺ
­ 6x y 5 0, ­ 3y 3 0,
­ y 1,
­ y 1,
® 6x 4y 2 0, ® 6x y 5 0, ®
®
6x
6
0,
¯ x 1.
¯
¯
¯
Ɉɬɜɟɬ: 0; (–1;1).
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
4
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
3
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
4
23
Ɇɨɞɭɥɶ "Ƚɟɨɦɟɬɪɢɹ"
24
Ɉɞɢɧ ɢɡ ɤɨɪɧɟɣ ɭɪɚɜɧɟɧɢɹ 5x 2x 3p
Ɋɟɲɟɧɢɟ.
ɉɨɞɫɬɚɜɢɦ ɢɡɜɟɫɬɧɵɣ ɤɨɪɟɧɶ ɜ ɭɪɚɜɧɟɧɢɟ:
ɨɬɧɨɫɢɬɟɥɶɧɨ p. Ɋɟɲɢɦ ɟɝɨ: 3p 3; p
5x 2 2x 3 0, ɨɬɤɭɞɚ
2r8
2r 44˜5˜3
,
x
10
10
2
22
0 ɪɚɜɟɧ 1. ɇɚɣɞɢɬɟ ɜɬɨɪɨɣ ɤɨɪɟɧɶ.
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 2
ɇɚɣɞɢɬɟ ɜɟɥɢɱɢɧɭ ɭɝɥɚ DOB, ɟɫɥɢ OE – ɛɢɫɫɟɤɬɪɢɫɚ ɭɝɥɚ AOC, OD –
ɛɢɫɫɟɤɬɪɢɫɚ ɭɝɥɚ COB.
5 2 3p 0. ɉɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ
1. ɉɨɞɫɬɚɜɢɦ p ɜ ɭɪɚɜɧɟɧɢɟ:
Ɋɟɲɟɧɢɟ.
x1
1, x2
0, 6.
Ɉɬɜɟɬ: 0, 6.
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
3
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
2
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
3
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
ӘCOȺ=2·64°=128°; ӘȼOC=180°–128°=52°; ӘDOB=52°:2=26°.
Ɉɬɜɟɬ: 26°.
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
Ɋɟɲɟɧɢɟ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
2
Ⱦɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɟɬɨɦ
1
ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
2
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 5
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 3
ȼ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɟ ABCD ɬɨɱɤɚ M — ɫɟɪɟɞɢɧɚ ɫɬɨɪɨɧɵ AB. ɂɡɜɟɫɬɧɨ, ɱɬɨ
MC MD. Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ɞɚɧɧɵɣ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦ — ɩɪɹɦɨɭɝɨɥɶɧɢɤ.
ɉɭɫɬɶ ɬɨɱɤɚ M – ɫɟɪɟɞɢɧɚ ɫɬɨɪɨɧɵ AB ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ABCD – ɪɚɜɧɨɭɞɚɥɟɧɚ ɨɬ ɟɝɨ
ɜɟɪɲɢɧ C ɢ D. Ɍɨɝɞɚ, ɬɪɟɭɝɨɥɶɧɢɤ CMD – ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ, ɩɨɷɬɨɦɭ ‘ MCD ‘ MDC .
ɉɨɫɤɨɥɶɤɭ
ɩɪɹɦɚɹ CD ɩɚɪɚɥɥɟɥɶɧɚ
ɫɬɨɪɨɧɟ
ɬɨ ‘ BMC ‘ MCD ɢ
AB,
‘ AMD ‘ MDC ɤɚɤ ɧɚɤɪɟɫɬ ɥɟɠɚɳɢɟ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, +BMC +AMD ɩɨ ɩɟɪɜɨɦɭ
ɩɪɢɡɧɚɤɭ ɪɚɜɟɧɫɬɜɚ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ( ‘ BMC ‘ AMD, AM BM , MC MD).
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 5
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 4
25
Ɂɧɚɱɢɬ, ‘ CBM ‘ DAM . ɂɯ ɫɭɦɦɚ ɪɚɜɧɚ 180q, ɬ.ɤ. ɷɬɨ ɞɜɚ ɭɝɥɚ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ,
ɩɪɢɥɟɠɚɳɢɟ ɤ ɨɞɧɨɣ ɫɬɨɪɨɧɟ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ‘ CBM ‘ DAM 90D. ɉɨ ɫɜɨɣɫɬɜɭ
ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ɭɝɥɵ BCD ɢ CDA ɬɚɤɠɟ ɩɪɹɦɵɟ. Ɂɧɚɱɢɬ, ABCD – ɩɪɹɦɨɭɝɨɥɶɧɢɤ.
Ʉɨɦɦɟɧɬɚɪɢɣ: Ɋɚɜɟɧɫɬɜɨ ɬɪɟɭɝɨɥɶɧɢɤɨɜ BMC ɢ AMD ɦɨɠɟɬ ɛɵɬɶ ɞɨɤɚɡɚɧɨ ɢɧɚɱɟ,
ɧɚɩɪɢɦɟɪ, ɩɨ ɬɪɟɬɶɟɦɭ ɩɪɢɡɧɚɤɭ ɪɚɜɟɧɫɬɜɚ ɬɪɟɭɝɨɥɶɧɢɤɨɜ.
Ⱦɪɭɝɨɟ ɜɨɡɦɨɠɧɨɟ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ:
ɉɭɫɬɶ ɬɨɱɤɚ O – ɫɟɪɟɞɢɧɚ CD. ɑɟɬɵɪɟɯɭɝɨɥɶɧɢɤ OMBC ɹɜɥɹɟɬɫɹ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɨɦ,
ɩɨɫɤɨɥɶɤɭ ɟɝɨ ɫɬɨɪɨɧɵ OC ɢ MB ɩɚɪɚɥɥɟɥɶɧɵ ɢ ɪɚɜɧɵ. Ɍɪɟɭɝɨɥɶɧɢɤ MCD –
ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ, ɩɨɷɬɨɦɭ OM – ɟɝɨ ɜɵɫɨɬɚ. Ɂɧɚɱɢɬ, OMBC – ɩɪɹɦɨɭɝɨɥɶɧɢɤ,
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɝɨɥ CBM – ɩɪɹɦɨɣ.
Ⱦɚɧɧɚɹ ɨɤɪɭɠɧɨɫɬɶ ɤɚɫɚɟɬɫɹ ɫɬɨɪɨɧɵ AC ɜ ɟɺ ɫɟɪɟɞɢɧɟ M ɢ ɩɪɨɞɨɥɠɟɧɢɣ ɫɬɨɪɨɧ BA ɢ
BC ɬɪɟɭɝɨɥɶɧɢɤɚ ABC.
ɉɭɫɬɶ O — ɰɟɧɬɪ ɷɬɨɣ ɨɤɪɭɠɧɨɫɬɢ, ɚ Q — ɰɟɧɬɪ ɨɤɪɭɠɧɨɫɬɢ, ɜɩɢɫɚɧɧɨɣ ɜ
ɬɪɟɭɝɨɥɶɧɢɤ ABC. ɍɝɨɥ OAQ – ɩɪɹɦɨɣ ɤɚɤ ɭɝɨɥ ɦɟɠɞɭ ɛɢɫɫɟɤɬɪɢɫɚɦɢ ɫɦɟɠɧɵɯ ɭɝɥɨɜ.
Ɍɪɟɭɝɨɥɶɧɢɤ OAQ – ɩɪɹɦɨɭɝɨɥɶɧɵɣ, AM – ɟɝɨ ɜɵɫɨɬɚ. ɂɡ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ɧɚɯɨɞɢɦ,
AM 2 9
4, 5.
ɱɬɨ AM 2 MQ ˜ MO. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, QM
2
OM
Ɉɬɜɟɬ: 4, 5.
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ
4
ɨɬɜɟɬ
ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɧɨ ɞɚɧɵ
3
ɧɟɩɨɥɧɵɟ ɨɛɴɹɫɧɟɧɢɹ ɢɥɢ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɨɲɢɛɤɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
4
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɜɟɪɧɨɟ, ɜɫɟ ɲɚɝɢ ɨɛɨɫɧɨɜɚɧɵ
3
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɫɨɞɟɪɠɢɬ ɧɟɬɨɱɧɨɫɬɢ ɢɥɢ ɩɪɨɛɟɥɵ, ɧɚɩɪɢɦɟɪ, ɨɬɫɭɬɫɬɜɭɸɬ
2
ɫɫɵɥɤɢ ɧɚ ɫɜɨɣɫɬɜɚ ɩɚɪɚɥɥɟɥɶɧɵɯ ɩɪɹɦɵɯ ɢɥɢ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
3
Ɉɫɧɨɜɚɧɢɟ AC ɪɚɜɧɨɛɟɞɪɟɧɧɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ABC ɪɚɜɧɨ 12. Ɉɤɪɭɠɧɨɫɬɶ
ɪɚɞɢɭɫɚ 8 ɫ ɰɟɧɬɪɨɦ ɜɧɟ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ɤɚɫɚɟɬɫɹ ɩɪɨɞɨɥɠɟɧɢɹ ɛɨɤɨɜɵɯ
ɫɬɨɪɨɧ ɬɪɟɭɝɨɥɶɧɢɤɚ ɢ ɤɚɫɚɟɬɫɹ ɨɫɧɨɜɚɧɢɹ AC ɜ ɟɝɨ ɫɟɪɟɞɢɧɟ. ɇɚɣɞɢɬɟ ɪɚɞɢɭɫ
ɨɤɪɭɠɧɨɫɬɢ, ɜɩɢɫɚɧɧɨɣ ɜ ɬɪɟɭɝɨɥɶɧɢɤ ABC.
Ɋɟɲɟɧɢɟ.
26
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 6
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 1
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɡɚɞɚɧɢɣ ɫ ɪɚɡɜɺɪɧɭɬɵɦ ɨɬɜɟɬɨɦ
ɍɩɪɨɫɬɢɬɟ ɜɵɪɚɠɟɧɢɟ
54
15 3 ˜
15 3
.
Ɋɟɲɟɧɢɟ.
54
15 3 ˜
Ɉɬɜɟɬ: 3.
54
15 3
( 15 3)( 15 3)
54
15 9
54
6
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 2
ɇɚɣɞɢɬɟ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ ɜɵɪɚɠɟɧɢɹ ɢ ɡɧɚɱɟɧɢɹ x ɢ y, ɩɪɢ ɤɨɬɨɪɵɯ ɨɧɨ
ɞɨɫɬɢɝɚɟɬɫɹ: 3x 4y 1 x 5y 6 .
Ɋɟɲɟɧɢɟ.
ɋɭɦɦɚ 3x 4y 1 x 5y 6 ɩɪɢɧɢɦɚɟɬ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ, ɪɚɜɧɨɟ 0, ɬɨɥɶɤɨ ɜ
ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɨɛɚ ɫɥɚɝɚɟɦɵɯ ɨɞɧɨɜɪɟɦɟɧɧɨ ɪɚɜɧɵ 0. ɉɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ
­ 3x 4y 1 0,
® x 5y 6 0.
¯
Ɋɟɲɢɦ ɟɺ
­ 3x 4y 1 0,
­ 19y 19 0, ­ y 1,
® 3x 15y 18 0; ® x 5y 6 0; ®
¯ x 1.
¯
¯
Ɉɬɜɟɬ: 0; (–1;1).
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
4
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
3
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
4
23
Ɇɨɞɭɥɶ "Ⱥɥɝɟɛɪɚ"
21
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 6
9
3.
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
2
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
1
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
2
Ʉɨɦɦɟɧɬɚɪɢɣ. Ɉɲɢɛɤɢ ɜ ɩɪɢɦɟɧɟɧɢɢ ɮɨɪɦɭɥ ɫɱɢɬɚɸɬɫɹ ɫɭɳɟɫɬɜɟɧɧɵɦɢ; ɩɪɢ ɢɯ
ɧɚɥɢɱɢɢ ɪɟɲɟɧɢɟ ɧɟ ɡɚɫɱɢɬɵɜɚɟɬɫɹ.
Ɉɞɢɧ ɢɡ ɤɨɪɧɟɣ ɭɪɚɜɧɟɧɢɹ 4x 2 x 3m 0 ɪɚɜɟɧ 1. ɇɚɣɞɢɬɟ ɜɬɨɪɨɣ ɤɨɪɟɧɶ.
Ɋɟɲɟɧɢɟ.
ɉɨɞɫɬɚɜɢɦ ɢɡɜɟɫɬɧɵɣ ɤɨɪɟɧɶ ɜ ɭɪɚɜɧɟɧɢɟ: 4 1 3m 0. ɉɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ
ɨɬɧɨɫɢɬɟɥɶɧɨ m. Ɋɟɲɢɦ ɟɝɨ: 3m 3; m 1. ɉɨɞɫɬɚɜɢɦ m ɜ ɭɪɚɜɧɟɧɢɟ:
4x 2 x 3 0, ɨɬɤɭɞɚ
3
1r7
1r 14˜3˜4
, x1 1, x2 .
x
4
8
8
3
Ɉɬɜɟɬ: .
4
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
3
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
2
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
3
22
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɨɞɭɥɶ "Ƚɟɨɦɟɬɪɢɹ"
24
ɇɚɣɞɢɬɟ ɜɟɥɢɱɢɧɭ ɭɝɥɚ AOE, ɟɫɥɢ OE – ɛɢɫɫɟɤɬɪɢɫɚ ɭɝɥɚ AOC, OD –
ɛɢɫɫɟɤɬɪɢɫɚ ɭɝɥɚ COB.
Ɋɟɲɟɧɢɟ.
ӘCOB=2·25°=50°; ӘAOC=180°–50°=130°; ӘAOE=130°:2=65°.
Ɉɬɜɟɬ: 65°.
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 6
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 3
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
Ɋɟɲɟɧɢɟ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
2
Ⱦɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɟɬɨɦ
1
ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
2
ȼ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɟ ABCD ɬɨɱɤɚ K — ɫɟɪɟɞɢɧɚ ɫɬɨɪɨɧɵ AB. ɂɡɜɟɫɬɧɨ, ɱɬɨ
KC KD. Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ɞɚɧɧɵɣ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦ — ɩɪɹɦɨɭɝɨɥɶɧɢɤ.
ɉɭɫɬɶ ɬɨɱɤɚ K – ɫɟɪɟɞɢɧɚ ɫɬɨɪɨɧɵ AB ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ABCD – ɪɚɜɧɨɭɞɚɥɟɧɚ ɨɬ ɟɝɨ
ɜɟɪɲɢɧ C ɢ D. Ɍɨɝɞɚ, ɬɪɟɭɝɨɥɶɧɢɤ CKD – ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ, ɩɨɷɬɨɦɭ ‘ KCD ‘ KDC .
ɉɨɫɤɨɥɶɤɭ ɩɪɹɦɚɹ CD ɩɚɪɚɥɥɟɥɶɧɚ ɫɬɨɪɨɧɟ AB, ɬɨ ‘ BKC ‘ KCD ɢ ‘ AKD ‘ KDC
ɤɚɤ ɧɚɤɪɟɫɬ ɥɟɠɚɳɢɟ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, +BKC +AKD ɩɨ ɩɟɪɜɨɦɭ ɩɪɢɡɧɚɤɭ ɪɚɜɟɧɫɬɜɚ
ɬɪɟɭɝɨɥɶɧɢɤɨɜ ( ‘ BKC ‘ AKD, AK BK , KC KD).
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 6
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 4
Ɉɫɧɨɜɚɧɢɟ AC ɪɚɜɧɨɛɟɞɪɟɧɧɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ABC ɪɚɜɧɨ 8. Ɉɤɪɭɠɧɨɫɬɶ
ɪɚɞɢɭɫɚ 6 ɫ ɰɟɧɬɪɨɦ ɜɧɟ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ɤɚɫɚɟɬɫɹ ɩɪɨɞɨɥɠɟɧɢɹ ɛɨɤɨɜɵɯ
ɫɬɨɪɨɧ ɬɪɟɭɝɨɥɶɧɢɤɚ ɢ ɤɚɫɚɟɬɫɹ ɨɫɧɨɜɚɧɢɹ AC ɜ ɟɝɨ ɫɟɪɟɞɢɧɟ. ɇɚɣɞɢɬɟ ɪɚɞɢɭɫ
ɨɤɪɭɠɧɨɫɬɢ, ɜɩɢɫɚɧɧɨɣ ɜ ɬɪɟɭɝɨɥɶɧɢɤ ABC.
Ɋɟɲɟɧɢɟ.
26
25
Ɂɧɚɱɢɬ, ‘ CBK ‘ DAK . ɂɯ ɫɭɦɦɚ ɪɚɜɧɚ 180 q, ɬ.ɤ. ɷɬɨ ɞɜɚ ɭɝɥɚ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ,
ɩɪɢɥɟɠɚɳɢɟ ɤ ɨɞɧɨɣ ɫɬɨɪɨɧɟ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ‘ CBK ‘ DAK 90D. ɉɨ ɫɜɨɣɫɬɜɭ
ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ɭɝɥɵ BCD ɢ CDA ɬɚɤɠɟ ɩɪɹɦɵɟ. Ɂɧɚɱɢɬ, ABCD – ɩɪɹɦɨɭɝɨɥɶɧɢɤ.
Ʉɨɦɦɟɧɬɚɪɢɣ: Ɋɚɜɟɧɫɬɜɨ ɬɪɟɭɝɨɥɶɧɢɤɨɜ BKC ɢ AKD ɦɨɠɟɬ ɛɵɬɶ ɞɨɤɚɡɚɧɨ ɢɧɚɱɟ,
ɧɚɩɪɢɦɟɪ, ɩɨ ɬɪɟɬɶɟɦɭ ɩɪɢɡɧɚɤɭ ɪɚɜɟɧɫɬɜɚ ɬɪɟɭɝɨɥɶɧɢɤɨɜ.
Ⱦɪɭɝɨɟ ɜɨɡɦɨɠɧɨɟ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ:
ɉɭɫɬɶ ɬɨɱɤɚ O – ɫɟɪɟɞɢɧɚ CD. ɑɟɬɵɪɟɯɭɝɨɥɶɧɢɤ OKBC ɹɜɥɹɟɬɫɹ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɨɦ,
ɩɨɫɤɨɥɶɤɭ ɟɝɨ ɫɬɨɪɨɧɵ OC ɢ KB ɩɚɪɚɥɥɟɥɶɧɵ ɢ ɪɚɜɧɵ. Ɍɪɟɭɝɨɥɶɧɢɤ KCD –
ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ, ɩɨɷɬɨɦɭ OK – ɟɝɨ ɜɵɫɨɬɚ. Ɂɧɚɱɢɬ, OKBC – ɩɪɹɦɨɭɝɨɥɶɧɢɤ,
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɝɨɥ CBK – ɩɪɹɦɨɣ.
Ⱦɚɧɧɚɹ ɨɤɪɭɠɧɨɫɬɶ ɤɚɫɚɟɬɫɹ ɫɬɨɪɨɧɵ AC ɜ ɟɺ ɫɟɪɟɞɢɧɟ M ɢ ɩɪɨɞɨɥɠɟɧɢɣ ɫɬɨɪɨɧ BA ɢ
BC ɬɪɟɭɝɨɥɶɧɢɤɚ ABC.
ɉɭɫɬɶ O – ɰɟɧɬɪ ɷɬɨɣ ɨɤɪɭɠɧɨɫɬɢ, ɚ Q – ɰɟɧɬɪ ɨɤɪɭɠɧɨɫɬɢ, ɜɩɢɫɚɧɧɨɣ ɜ ɬɪɟɭɝɨɥɶɧɢɤ
ABC. ɍɝɨɥ OAQ – ɩɪɹɦɨɣ ɤɚɤ ɭɝɨɥ ɦɟɠɞɭ ɛɢɫɫɟɤɬɪɢɫɚɦɢ ɫɦɟɠɧɵɯ ɭɝɥɨɜ. Ɍɪɟɭɝɨɥɶɧɢɤ
OAQ – ɩɪɹɦɨɭɝɨɥɶɧɵɣ, AM – ɟɝɨ ɜɵɫɨɬɚ. ɂɡ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ɧɚɯɨɞɢɦ, ɱɬɨ
AM 2 8
.
AM 2 MQ ˜ MO. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, QM
OM
3
8
Ɉɬɜɟɬ: .
3
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ
4
ɨɬɜɟɬ
ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɧɨ ɞɚɧɵ
3
ɧɟɩɨɥɧɵɟ ɨɛɴɹɫɧɟɧɢɹ ɢɥɢ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɨɲɢɛɤɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
4
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɜɟɪɧɨɟ, ɜɫɟ ɲɚɝɢ ɨɛɨɫɧɨɜɚɧɵ
3
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɫɨɞɟɪɠɢɬ ɧɟɬɨɱɧɨɫɬɢ ɢɥɢ ɩɪɨɛɟɥɵ, ɧɚɩɪɢɦɟɪ, ɨɬɫɭɬɫɬɜɭɸɬ
2
ɫɫɵɥɤɢ ɧɚ ɫɜɨɣɫɬɜɚ ɩɚɪɚɥɥɟɥɶɧɵɯ ɩɪɹɦɵɯ ɢɥɢ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
3
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 7
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 1
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɡɚɞɚɧɢɣ ɫ ɪɚɡɜɺɪɧɭɬɵɦ ɨɬɜɟɬɨɦ
Ɇɨɞɭɥɶ "Ⱥɥɝɟɛɪɚ"
21
ɍɩɪɨɫɬɢɬɟ ɜɵɪɚɠɟɧɢɟ
15 3 ˜
24
15 3
.
Ɋɟɲɟɧɢɟ.
( 15 3)( 15 3)
1
1
15 3 ˜
15 3
15 9
6
.
24
24
24
24
4
2
1
Ɉɬɜɟɬ: .
2
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
2
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
1
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
2
Ʉɨɦɦɟɧɬɚɪɢɣ. Ɉɲɢɛɤɢ ɜ ɩɪɢɦɟɧɟɧɢɢ ɮɨɪɦɭɥ ɫɱɢɬɚɸɬɫɹ ɫɭɳɟɫɬɜɟɧɧɵɦɢ; ɩɪɢ ɢɯ
ɧɚɥɢɱɢɢ ɪɟɲɟɧɢɟ ɧɟ ɡɚɫɱɢɬɵɜɚɟɬɫɹ.
Ɉɞɢɧ ɢɡ ɤɨɪɧɟɣ ɭɪɚɜɧɟɧɢɹ 5x 2 7x 2m 0 ɪɚɜɟɧ –1. ɇɚɣɞɢɬɟ ɜɬɨɪɨɣ ɤɨɪɟɧɶ.
Ɋɟɲɟɧɢɟ.
ɉɨɞɫɬɚɜɢɦ ɢɡɜɟɫɬɧɵɣ ɤɨɪɟɧɶ ɜ ɭɪɚɜɧɟɧɢɟ: 5 7 2m 0. ɉɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ
ɨɬɧɨɫɢɬɟɥɶɧɨ m. Ɋɟɲɢɦ ɟɝɨ: 2m 2; m 1. ɉɨɞɫɬɚɜɢɦ m ɜ ɭɪɚɜɧɟɧɢɟ:
5x 2 7x 2 0, ɨɬɤɭɞɚ
7 r 3
7 r 49 4 ˜ 5 ˜ 2
x
, x1 1, x2 0, 4.
10
10
Ɉɬɜɟɬ: 0, 4.
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
3
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
2
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
3
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 7
ɇɚɣɞɢɬɟ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ ɜɵɪɚɠɟɧɢɹ ɢ ɡɧɚɱɟɧɢɹ x ɢ y, ɩɪɢ ɤɨɬɨɪɵɯ ɨɧɨ
ɞɨɫɬɢɝɚɟɬɫɹ 6x 5y 7 2x 3y 1 .
Ɋɟɲɟɧɢɟ.
ɋɭɦɦɚ 6x 5y 7 2x 3y 1 ɩɪɢɧɢɦɚɟɬ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ, ɪɚɜɧɨɟ 0, ɬɨɥɶɤɨ ɜ
ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɨɛɚ ɫɥɚɝɚɟɦɵɯ ɨɞɧɨɜɪɟɦɟɧɧɨ ɪɚɜɧɵ 0. ɉɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ
­ 6x 5y 7 0,
® 2x 3y 1 0.
¯
Ɋɟɲɢɦ ɟɺ:
­ 6x 5y 7 0, ­ 4y 4 0,
­ y 1,
­ y 1,
® 6x 9y 3 0; ® 6x 9y 3 0; ®
®
6x
12
0;
¯ x 2.
¯
¯
¯
Ɉɬɜɟɬ: 0; (–2;1).
23
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
4
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
3
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
4
Ɇɨɞɭɥɶ "Ƚɟɨɦɟɬɪɢɹ"
22
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 2
24
ɇɚɣɞɢɬɟ ɜɟɥɢɱɢɧɭ ɭɝɥɚ COE, ɟɫɥɢ OE – ɛɢɫɫɟɤɬɪɢɫɚ ɭɝɥɚ AOC, OD –
ɛɢɫɫɟɤɬɪɢɫɚ ɭɝɥɚ COB.
Ɋɟɲɟɧɢɟ.
ӘCOB=2·35°=70°; ӘAOC=180°–70°=110°; ӘɋOE=110°:2=55°.
Ɉɬɜɟɬ: 55°.
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 7
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 3
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 7
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 4
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
Ɋɟɲɟɧɢɟ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
2
Ⱦɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɟɬɨɦ
1
ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
2
ɉɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɭɝɥɵ ɱɟɬɵɪɟɯɭɝɨɥɶɧɢɤɚ ɩɨɩɚɪɧɨ ɪɚɜɧɵ. Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ɷɬɨɬ
ɱɟɬɵɪɟɯɭɝɨɥɶɧɢɤ – ɩɚɪɚɥɥɟɥɨɝɪɚɦɦ.
ɉɭɫɬɶ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɭɝɥɵ A ɢ C ɱɟɬɵɪɟɯɭɝɨɥɶɧɢɤɚ ABCD ɪɚɜɧɵ Į , ɚ
ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɭɝɥɵ B ɢ D ɪɚɜɧɵ ȕ. ɉɨɫɤɨɥɶɤɭ ɫɭɦɦɚ ɭɝɥɨɜ ɥɸɛɨɝɨ
ɱɟɬɵɪɟɯɭɝɨɥɶɧɢɤɚ ɪɚɜɧɚ 360q, ɬɨ 2Į 2ȕ 360q. Ɂɧɚɱɢɬ, Į ȕ 180q. Ɍɚɤ ɤɚɤ ɫɭɦɦɚ
ɜɧɭɬɪɟɧɧɢɯ ɨɞɧɨɫɬɨɪɨɧɧɢɯ ɭɝɥɨɜ ɩɪɢ ɫɟɤɭɳɟɣ ɪɚɜɧɚ 180D,ɬɨ ɩɨ ɩɪɢɡɧɚɤɭ
ɩɚɪɚɥɥɟɥɶɧɵɯ ɩɪɹɦɵɯ AB ɩɚɪɚɥɥɟɥɶɧɚ CD, BC ɩɚɪɚɥɥɟɥɶɧɚ AD. Ɂɧɚɱɢɬ,
ɱɟɬɵɪɺɯɭɝɨɥɶɧɢɤ ABCD ɹɜɥɹɟɬɫɹ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɨɦ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ.
25
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɜɟɪɧɨɟ, ɜɫɟ ɲɚɝɢ ɨɛɨɫɧɨɜɚɧɵ
3
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɫɨɞɟɪɠɢɬ ɧɟɬɨɱɧɨɫɬɢ ɢɥɢ ɩɪɨɛɟɥɵ, ɧɚɩɪɢɦɟɪ, ɨɬɫɭɬɫɬɜɭɸɬ
2
ɫɫɵɥɤɢ ɧɚ ɫɜɨɣɫɬɜɚ ɩɚɪɚɥɥɟɥɶɧɵɯ ɩɪɹɦɵɯ ɢɥɢ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
3
Ⱦɚɧɧɚɹ ɨɤɪɭɠɧɨɫɬɶ ɤɚɫɚɟɬɫɹ ɫɬɨɪɨɧɵ AC ɜ ɟɺ ɫɟɪɟɞɢɧɟ M ɢ ɩɪɨɞɨɥɠɟɧɢɣ ɫɬɨɪɨɧ BA ɢ
BC ɬɪɟɭɝɨɥɶɧɢɤɚ ABC.
ɉɭɫɬɶ O — ɰɟɧɬɪ ɷɬɨɣ ɨɤɪɭɠɧɨɫɬɢ, ɚ Q — ɰɟɧɬɪ ɨɤɪɭɠɧɨɫɬɢ, ɜɩɢɫɚɧɧɨɣ ɜ
ɬɪɟɭɝɨɥɶɧɢɤ ABC. ɍɝɨɥ OAQ – ɩɪɹɦɨɣ ɤɚɤ ɭɝɨɥ ɦɟɠɞɭ ɛɢɫɫɟɤɬɪɢɫɚɦɢ ɫɦɟɠɧɵɯ ɭɝɥɨɜ.
Ɍɪɟɭɝɨɥɶɧɢɤ OAQ – ɩɪɹɦɨɭɝɨɥɶɧɵɣ, AM – ɟɝɨ ɜɵɫɨɬɚ. ɂɡ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ɧɚɯɨɞɢɦ,
AM 2 9
ɱɬɨ AM 2 MQ ˜ MO. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, QM
1, 8.
5
OM
Ɉɬɜɟɬ: 1, 8.
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ
4
ɨɬɜɟɬ
ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɧɨ ɞɚɧɵ
3
ɧɟɩɨɥɧɵɟ ɨɛɴɹɫɧɟɧɢɹ ɢɥɢ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɨɲɢɛɤɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
4
Ɉɫɧɨɜɚɧɢɟ AC ɪɚɜɧɨɛɟɞɪɟɧɧɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ABC ɪɚɜɧɨ 6. Ɉɤɪɭɠɧɨɫɬɶ
ɪɚɞɢɭɫɚ 5 ɫ ɰɟɧɬɪɨɦ ɜɧɟ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ɤɚɫɚɟɬɫɹ ɩɪɨɞɨɥɠɟɧɢɹ ɛɨɤɨɜɵɯ
ɫɬɨɪɨɧ ɬɪɟɭɝɨɥɶɧɢɤɚ ɢ ɤɚɫɚɟɬɫɹ ɨɫɧɨɜɚɧɢɹ AC ɜ ɟɝɨ ɫɟɪɟɞɢɧɟ. ɇɚɣɞɢɬɟ ɪɚɞɢɭɫ
ɨɤɪɭɠɧɨɫɬɢ, ɜɩɢɫɚɧɧɨɣ ɜ ɬɪɟɭɝɨɥɶɧɢɤ ABC.
Ɋɟɲɟɧɢɟ.
26
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 8
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 1
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɡɚɞɚɧɢɣ ɫ ɪɚɡɜɺɪɧɭɬɵɦ ɨɬɜɟɬɨɦ
ɍɩɪɨɫɬɢɬɟ ɜɵɪɚɠɟɧɢɟ
54
31 5 ˜
31 5
.
Ɋɟɲɟɧɢɟ.
54
31 5 ˜
Ɉɬɜɟɬ: 3.
54
31 5
( 31 5)( 31 5)
54
31 25
54
6
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 2
ɇɚɣɞɢɬɟ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ ɜɵɪɚɠɟɧɢɹ ɢ ɡɧɚɱɟɧɢɹ x ɢ y, ɩɪɢ ɤɨɬɨɪɵɯ ɨɧɨ
ɞɨɫɬɢɝɚɟɬɫɹ: 3x 4y 2 x 5y 3 .
Ɋɟɲɟɧɢɟ.
ɋɭɦɦɚ 3x 4y 2 x 5y 3 ɩɪɢɧɢɦɚɟɬ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ, ɪɚɜɧɨɟ 0, ɬɨɥɶɤɨ ɜ
ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɨɛɚ ɫɥɚɝɚɟɦɵɯ ɨɞɧɨɜɪɟɦɟɧɧɨ ɪɚɜɧɵ 0. ɉɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ
­ 3x 4y 2 0,
® x 5y 3 0.
¯
Ɋɟɲɢɦ ɟɺ:
­ 3x 4y 2 0, ­ 11y 11 0, ­ y 1,
® 3x 15y 9 0, ® x 5y 3 0, ®
¯ x 2.
¯
¯
Ɉɬɜɟɬ: 0; (2;1).
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
4
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
3
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
4
23
Ɇɨɞɭɥɶ "Ⱥɥɝɟɛɪɚ"
21
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 8
9
3.
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
2
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
1
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
2
Ʉɨɦɦɟɧɬɚɪɢɣ. Ɉɲɢɛɤɢ ɜ ɩɪɢɦɟɧɟɧɢɢ ɮɨɪɦɭɥ ɫɱɢɬɚɸɬɫɹ ɫɭɳɟɫɬɜɟɧɧɵɦɢ; ɩɪɢ ɢɯ
ɧɚɥɢɱɢɢ ɪɟɲɟɧɢɟ ɧɟ ɡɚɫɱɢɬɵɜɚɟɬɫɹ.
Ɉɞɢɧ ɢɡ ɤɨɪɧɟɣ ɭɪɚɜɧɟɧɢɹ 3x 2 5x 2m 0 ɪɚɜɟɧ 1. ɇɚɣɞɢɬɟ ɜɬɨɪɨɣ ɤɨɪɟɧɶ.
Ɋɟɲɟɧɢɟ.
ɉɨɞɫɬɚɜɢɦ ɢɡɜɟɫɬɧɵɣ ɤɨɪɟɧɶ ɜ ɭɪɚɜɧɟɧɢɟ: 3 5 2m 0. ɉɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ
ɨɬɧɨɫɢɬɟɥɶɧɨ m.Ɋɟɲɢɦ ɟɝɨ: 2m 2; m 1. ɉɨɞɫɬɚɜɢɦ m ɜ ɭɪɚɜɧɟɧɢɟ: 3x 2 5x 2 0,
ɨɬɤɭɞɚ
2
5 r 25 4 ˜ 3 ˜ 2
5 r 1
, x1 1, x2 .
x
6
6
3
2
Ɉɬɜɟɬ: .
3
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ȼɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɟɧɵ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
3
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ
2
ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɺɬɨɦ ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
3
Ɇɨɞɭɥɶ "Ƚɟɨɦɟɬɪɢɹ"
22
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
24
ɇɚɣɞɢɬɟ ɜɟɥɢɱɢɧɭ ɭɝɥɚ COE, ɟɫɥɢ OE – ɛɢɫɫɟɤɬɪɢɫɚ ɭɝɥɚ AOC, OD –
ɛɢɫɫɟɤɬɪɢɫɚ ɭɝɥɚ COB.
Ɋɟɲɟɧɢɟ.
ӘCOB=2·32°=64°; ӘAOC=180°–64°=116°; ӘɋOE=116°:2=58°.
Ɉɬɜɟɬ: 58°.
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
Ɋɟɲɟɧɢɟ ɜɟɪɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ ɨɬɜɟɬ
2
Ⱦɨɩɭɳɟɧɚ ɨɞɧɚ ɨɲɢɛɤɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɢɥɢ ɨɩɢɫɤɚ, ɫ ɟɺ ɭɱɟɬɨɦ
1
ɪɟɲɟɧɢɟ ɞɨɜɟɞɟɧɨ ɞɨ ɤɨɧɰɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
2
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 8
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 3
ɋɟɪɟɞɢɧɚ ɫɬɨɪɨɧɵ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ɪɚɜɧɨɭɞɚɥɟɧɚ ɨɬ ɤɨɧɰɨɜ ɟɝɨ
ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɣ ɫɬɨɪɨɧɵ. Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ɞɚɧɧɵɣ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦ –
ɩɪɹɦɨɭɝɨɥɶɧɢɤ.
ɉɭɫɬɶ ɬɨɱɤɚ O – ɫɟɪɟɞɢɧɚ ɫɬɨɪɨɧɵ BC ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ABCD – ɪɚɜɧɨɭɞɚɥɟɧɚ ɨɬ ɟɝɨ
ɜɟɪɲɢɧ A ɢ D.Ɍɨɝɞɚ ɬɪɟɭɝɨɥɶɧɢɤ AOD ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ, ɩɨɷɬɨɦɭ ‘ AOD ‘ ODA.
ɉɨɫɤɨɥɶɤɭ ɩɪɹɦɚɹ ȼɋ ɩɚɪɚɥɥɟɥɶɧɚ ɫɬɨɪɨɧɟ AD, ɬɨ ɭɝɥɵ BOA ɢ COD ɪɚɜɧɵ ɭɤɚɡɚɧɧɵɦ
ɭɝɥɚɦ ɤɚɤ ɧɚɤɪɟɫɬ ɥɟɠɚɳɢɟ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, +BOA +COD ɩɨ ɩɟɪɜɨɦɭ ɩɪɢɡɧɚɤɭ
ɪɚɜɟɧɫɬɜɚ ɬɪɟɭɝɨɥɶɧɢɤɨɜ. Ɂɧɚɱɢɬ, ‘ ABO ‘ ODA.ɉɭɫɬɶ ɢɯ ɜɟɥɢɱɢɧɚ ɪɚɜɧɚ Į . ɉɪɹɦɵɟ
AB ɢ CD ɩɚɪɚɥɥɟɥɶɧɵ, ɩɨɷɬɨɦɭ Į Į 180q, ɬ.ɟ. Į 90q. ɉɨ ɫɜɨɣɫɬɜɭ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ
ɭɝɥɵ BAD ɢ CDA ɬɚɤɠɟ ɩɪɹɦɵɟ. Ɂɧɚɱɢɬ, ABCD – ɩɪɹɦɨɭɝɨɥɶɧɢɤ.
25
Ɇɚɬɟɦɚɬɢɤɚ. 9 ɤɥɚɫɫ. ȼɚɪɢɚɧɬ 8
ȼɢɞɟɨɪɚɡɛɨɪ ɧɚ ɫɚɣɬɟ www.statgrad.cde.ru 4
ɉɭɫɬɶ O — ɰɟɧɬɪ ɷɬɨɣ ɨɤɪɭɠɧɨɫɬɢ, ɚ Q — ɰɟɧɬɪ ɨɤɪɭɠɧɨɫɬɢ, ɜɩɢɫɚɧɧɨɣ ɜ
ɬɪɟɭɝɨɥɶɧɢɤ ABC. ɍɝɨɥ OAQ – ɩɪɹɦɨɣ ɤɚɤ ɭɝɨɥ ɦɟɠɞɭ ɛɢɫɫɟɤɬɪɢɫɚɦɢ ɫɦɟɠɧɵɯ ɭɝɥɨɜ.
Ɍɪɟɭɝɨɥɶɧɢɤ OAQ – ɩɪɹɦɨɭɝɨɥɶɧɵɣ, AM – ɟɝɨ ɜɵɫɨɬɚ. ɂɡ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ɧɚɯɨɞɢɦ,
AM 2 10
ɱɬɨ AM 2 MQ ˜ MO. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, QM
.
3
OM
10
.
Ɉɬɜɟɬ:
3
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɩɨɥɭɱɟɧ ɜɟɪɧɵɣ
4
ɨɬɜɟɬ
ɏɨɞ ɪɟɲɟɧɢɹ ɜɟɪɧɵɣ, ɜɫɟ ɟɝɨ ɲɚɝɢ ɜɵɩɨɥɧɟɧɵ ɩɪɚɜɢɥɶɧɨ, ɧɨ ɞɚɧɵ
3
ɧɟɩɨɥɧɵɟ ɨɛɴɹɫɧɟɧɢɹ ɢɥɢ ɞɨɩɭɳɟɧɚ ɨɞɧɚ ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɨɲɢɛɤɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
4
Ʉɪɢɬɟɪɢɢ ɨɰɟɧɢɜɚɧɢɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ
Ȼɚɥɥɵ
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɜɟɪɧɨɟ, ɜɫɟ ɲɚɝɢ ɨɛɨɫɧɨɜɚɧɵ
3
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɫɨɞɟɪɠɢɬ ɧɟɬɨɱɧɨɫɬɢ ɢɥɢ ɩɪɨɛɟɥɵ, ɧɚɩɪɢɦɟɪ, ɨɬɫɭɬɫɬɜɭɸɬ
2
ɫɫɵɥɤɢ ɧɚ ɫɜɨɣɫɬɜɚ ɩɚɪɚɥɥɟɥɶɧɵɯ ɩɪɹɦɵɯ ɢɥɢ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ
Ⱦɪɭɝɢɟ ɫɥɭɱɚɢ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɤɚɡɚɧɧɵɦ ɤɪɢɬɟɪɢɹɦ
0
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɛɚɥɥ
3
Ɉɫɧɨɜɚɧɢɟ AC ɪɚɜɧɨɛɟɞɪɟɧɧɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ABC ɪɚɜɧɨ 10. Ɉɤɪɭɠɧɨɫɬɶ
ɪɚɞɢɭɫɚ 7, 5 ɫ ɰɟɧɬɪɨɦ ɜɧɟ ɷɬɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ɤɚɫɚɟɬɫɹ ɩɪɨɞɨɥɠɟɧɢɹ ɛɨɤɨɜɵɯ
ɫɬɨɪɨɧ ɬɪɟɭɝɨɥɶɧɢɤɚ ɢ ɤɚɫɚɟɬɫɹ ɨɫɧɨɜɚɧɢɹ AC ɜ ɟɝɨ ɫɟɪɟɞɢɧɟ. ɇɚɣɞɢɬɟ ɪɚɞɢɭɫ
ɨɤɪɭɠɧɨɫɬɢ, ɜɩɢɫɚɧɧɨɣ ɜ ɬɪɟɭɝɨɥɶɧɢɤ ABC.
Ɋɟɲɟɧɢɟ.
26
Ⱦɚɧɧɚɹ ɨɤɪɭɠɧɨɫɬɶ ɤɚɫɚɟɬɫɹ ɫɬɨɪɨɧɵ AC ɜ ɟɺ ɫɟɪɟɞɢɧɟ M ɢ ɩɪɨɞɨɥɠɟɧɢɣ ɫɬɨɪɨɧ BA ɢ
BC ɬɪɟɭɝɨɥɶɧɢɤɚ ABC.
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
© ɆɂɈɈ 2012 ɝ. ɉɭɛɥɢɤɚɰɢɹ ɜ ɂɧɬɟɪɧɟɬɟ ɢɥɢ ɩɟɱɚɬɧɵɯ ɢɡɞɚɧɢɹɯ ɛɟɡ ɩɢɫɶɦɟɧɧɨɝɨ ɫɨɝɥɚɫɢɹ ɆɂɈɈ ɡɚɩɪɟɳɟɧɚ
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