РЕШЕНИЕ:



S∆ABK + S∆AKM = S∆ / 2 ⇒ S∆ABK = S∆ / 2 – S∆AKM

BK:KM=10:9 ⇒
S∆ABK = 10
S∆AKM__9

S∆ / 2 – S∆AKM = 10
S∆AKM__9

9S∆ – 18 S∆AKM = 20 S∆AKM

38 S∆AKM = 9S∆

S∆AKM = 9S∆ / 38

∆AKM ∞ ∆NKB

AM = KM
BN__BK

x = 9
BN__10

BN = 10x / 9

∆ACP ∞ ∆NBP

AC = PC
BN__BP

2x__ = PC
10x / 9__BP

9 = PC
5 __BP

S∆ABP = BP
S∆APC__PC

S∆ABP = 9
S∆APC__5

S∆ABP + S∆APC = S∆ ⇒ S∆ABP = S∆ – S∆APC

S∆ – S∆APC = 9
S∆APC______5

5 S∆ – 5 S∆APC = 9 S∆APC

20 S∆APC = 5 S∆

S∆APC = S∆ / 4

S KPCM = S∆APC – S∆AKM = S∆ / 4– 9S∆ / 38 = 1 S∆ / 76

S KPCM / S∆= 1 S∆ / 76 : S∆ = 1/76

Ответ: 1/76

РЕШЕНИЕ:



S∆ABK + S∆AKM = S∆ / 2 ⇒ S∆ABK = S∆ / 2 – S∆AKM

BK:KM=7:2 ⇒
S∆ABK = 7
S∆AKM__2

S∆ / 2 – S∆AKM = 7
S∆AKM__2

S∆ – 2 S∆AKM = 7 S∆AKM

9 S∆AKM = S∆

S∆AKM = S∆ / 9

∆AKM ∞ ∆NKB

AM = KM
BN__BK

x = 2
BN__7

BN = 7x / 2

∆ACP ∞ ∆NBP

AC = PC
BN__BP

2x__ = PC
7x / 2__BP

4 = PC
7 __BP

S∆ABP = BP
S∆APC__PC

S∆ABP = 7
S∆APC__4

S∆ABP + S∆APC = S∆ ⇒ S∆ABP = S∆ – S∆APC

S∆ – S∆APC = 7
S∆APC______4

4 S∆ – 4 S∆APC = 7 S∆APC

11 S∆APC = 4 S∆

S∆APC = 4 S∆ / 11

S KPCM = S∆APC – S∆AKM = 4 S∆ / 11 – S∆ / 9 = 25S∆ / 99

S KPCM / S∆= 25S∆ / 99 : S∆ = 25/99

Ответ: 25/99

РЕШЕНИЕ:



S∆ABK + S∆AKM = S∆ / 2 ⇒ S∆ABK = S∆ / 2 – S∆AKM

BK:KM=4:9 ⇒
S∆ABK = 4
S∆AKM__9

S∆ / 2 – S∆AKM = 4
S∆AKM__9

9S∆ – 18 S∆AKM = 8 S∆AKM

26 S∆AKM = 9S∆

S∆AKM = 9S∆ / 26

∆AKM ∞ ∆NKB

AM = KM
BN__BK

x = 9
BN__4

BN = 4x / 9

∆ACP ∞ ∆NBP

AC = PC
BN__BP

2x__ = PC
4x / 9__BP

9 = PC
2 __BP

S∆ABP = BP
S∆APC__PC

S∆ABP = 2
S∆APC__9

S∆ABP + S∆APC = S∆ ⇒ S∆ABP = S∆ – S∆APC

S∆ – S∆APC = 2
S∆APC______9

9 S∆ – 9 S∆APC = 2 S∆APC

11 S∆APC = 9 S∆

S∆APC = 9 S∆ / 11

S KPCM = S∆APC – S∆AKM = 9 S∆ / 11 – 9S∆ / 26 = 135S∆ / 286

S∆AKM / S KPCM = 9S∆ / 26 : 135S∆ / 286 = 9/26 ∙ 286/135 = 11/15

Ответ: 11/15

РЕШЕНИЕ:


BK:KM=7:3

BK = 7x
KM = 3x
BM = 3x + 7x = 10 x

S _{∆ ABK} : S _{∆ ABC} = 7x / (2 ∙ 10x) = 7 / 20

Ответ: 7 : 20

РЕШЕНИЕ:



S∆ABK + S∆AKM = S∆ / 2 ⇒ S∆ABK = S∆ / 2 – S∆AKM

BK:KM=7:2 ⇒
S∆ABK = 7
S∆AKM__2

S∆ / 2 – S∆AKM = 7
S∆AKM__2

S∆ – 2 S∆AKM = 7 S∆AKM

9 S∆AKM = S∆

S∆AKM = S∆ / 9

S∆AKM : S∆ = S∆/9 : S∆ = 1/9

Ответ: 1/9

РЕШЕНИЕ:



S∆ABK + S∆AKM = S∆ / 2 ⇒ S∆ABK = S∆ / 2 – S∆AKM

BK:KM=2:11 ⇒
S∆ABK = 2
S∆AKM__11

S∆ / 2 – S∆AKM = 2
S∆AKM__11

11S∆ – 22 S∆AKM = 4 S∆AKM

26 S∆AKM = 11S∆

S∆AKM = 11S∆ / 26

∆AKM ∞ ∆NKB

AM = KM
BN__BK

x = 11
BN__2

BN = 2x / 11

∆ACP ∞ ∆NBP

AC = PC
BN__BP

2x__ = PC
2x / 11__BP

11 = PC
1 __BP

S∆ABP = BP
S∆APC__PC

S∆ABP = 1
S∆APC__11

S∆ABP + S∆APC = S∆ ⇒ S∆ABP = S∆ – S∆APC

S∆ – S∆APC = 1
S∆APC______11

11 S∆ – 11 S∆APC = 1 S∆APC

12 S∆APC = 11S∆

S∆APC = 11S∆ / 12

S KPCM = S∆APC – S∆AKM = 11S∆ / 12 – 11S∆ / 26 = 77 S∆ / 156

S∆BPK = S∆ / 2 – SKPCM = S∆ / 2 – 77 S∆ / 156 = S∆ / 156

S∆BKP / S ∆AKM = S∆ / 156 : 11S∆ / 26 = 1/156 ∙ 26/11 = 1/66

Ответ: 1/66