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Îòæå, êîîðäèíàòè òî÷êè Ñ (õ,ó), ÿêà ä³ëèòü â³äð³çîê À ó<br />
â³äíîøåíí³ λ (â³äë³ê â³ä À), âèçíà÷àþòüñÿ ôîðìóëàìè:<br />
x1 +λx2<br />
x = ,<br />
1 +λ<br />
y1 +λy<br />
(4.1.6)<br />
2<br />
y = .<br />
1 +λ<br />
Ðîçãëÿíåìî îêðåìèé âèïàäîê. Íåõàé òî÷êà Ñ ä³ëèòü â³äð³çîê<br />
À ïîïîëàì. Òîä³ ÀÑ = Ñ ³ λ=1. Ïîçíà÷èâøè êîîðäèíàòè<br />
ñåðåäèíè â³äð³çêà ÷åðåç õ ñ ³ ó ñ , îäåðæèìî íà ï³äñòàâ³<br />
ôîðìóë (4.1.6) òàê³<br />
x1 + x2<br />
xc<br />
= ,<br />
2<br />
y1 + y<br />
(4.1.7)<br />
2<br />
yc<br />
= ,<br />
2<br />
ç ÿêèõ âèäíî êîîðäèíàòè ñåðåäèíè â³äð³çêà äîð³âíþþòü<br />
ï³âñóìàì â³äïîâ³äíèõ êîîðäèíàò éîãî ê³íö³â.<br />
Ïðèì³òêà. Ïðè âèâåäåíí³ ôîðìóë (4.1.6) — (4.1.7) ìè<br />
ïðèïóñêàëè, ùî ê³íö³ À ³  â³äð³çêà À ëåæàòü ó ïåðøîìó<br />
êâàäðàíò³. Ëåãêî äîâåñòè, ùî ôîðìóëè (4.1.6) — (4.1.7)<br />
áóäóòü ñïðàâåäëèâ³ é ó òèõ âèïàäêàõ, êîëè ê³íö³ â³äð³çêà<br />
À ëåæàòü â ³íøèõ êâàäðàíòàõ.<br />
Ïðèêëàä 4.1.1. Îá÷èñëèòè êîîðäèíàòè òî÷êè Ñ(õ, ó), ùî<br />
ä³ëèòü â³äð³çîê AB ì³æ òî÷êàìè À (5, 7) ³  (3, –5) ó â³äíîøåíí³<br />
2<br />
AC<br />
CB = .<br />
Ðîçâ’ÿçàííÿ. Ó öüîìó âèïàäêó λ = 2. Îòæå,<br />
x<br />
c<br />
5+ 2⋅ 3 11 7+ 2( −5)<br />
= = , yc<br />
= = − 1.<br />
3 3 3<br />
4.1.4. Ïëîùà òðèêóòíèêà<br />
Íåõàé ïîòð³áíî çíàéòè ïëîùó S òðèêóòíèêà ÀÂÑ<br />
(ðèñ. 4.6) ç âåðøèíàìè A(x 1 , y 1 ), B(x 2 , y 2 ), C(x 3 , y 3 ). Íå îáìåæóþ÷è<br />
çàãàëüíîñò³, ïðèïóñòèìî, ùî âåðøèíè òðèêóòíèêà<br />
çíàõîäÿòüñÿ ó 1-ìó êâàäðàíò³. ×åðåç A′(x 1 ,0), B′(x 2 ,0),<br />
C′(x 3 ,0), A′′(0,y 1 ), B′′(0,y 2 ), C′′(0,y 3 ) â³äïîâ³äíî ïîçíà÷èìî ïðîåêö³¿<br />
òî÷îê A, B, C íà â³ñ³ êîîðäèíàò Ox ³ Oy.<br />
Íåõàé ÀÂ = c, ÀÑ = b, à êóòè, ÿê³ óòâîðåí³ öèìè ñòîðîíàìè<br />
ç â³ññþ Îõ, â³äïîâ³äíî ð³âí³ α i β (ðèñ. 4.6). Ç öüîãî ðèñóíêà<br />
áóäåìî ìàòè òàê³ ñï³ââ³äíîøåííÿ:<br />
AB ′ ′ = ccos α = x2 − x1, AC ′ ′ = bcos β = x3 −x1,<br />
AB ′′ ′′ = csin α = y − y, AC ′′ ′′ = bsin β = y − y.<br />
2 1 3 1<br />
(4.1.8)<br />
Ðèñ. 4.6<br />
Íåõàé j=Ð CAB . Òîä³ î÷åâèäíî (ðèñ. 4.6), ùî ϕ=β−α ³<br />
çà â³äîìîþ ôîðìóëîþ ç òðèãîíîìåò𳿠(äèâ. äîä. 2) áóäåìî<br />
ìàòè<br />
1 1 1<br />
S = bcsin ϕ = bcsin( β−α ) = bc(sin βcos α −cosβsin α ) .<br />
2 2 2<br />
Âðàõîâóþ÷è ôîðìóëè (4.1.8), îòðèìàºìî, ùî<br />
1<br />
S = [( y3 −y1 )( x2 −x1 ) − ( x3 −x1 )( y2 − y1<br />
)] . (4.1.9)<br />
2<br />
92 93