Task/24841973 ---.---.---.---.--- знайти площу трикутника якщо його висоти дорівнюють 24, 30, 40 см --------------------------------- По формуле Герона S =√p(p-a)(p-b)(p-c), где p =(a+b+c)/2_полупериметр
обозначаем h(a)=h₁ ; h(b)=h₂ ; h(c) =h₃ S =a*h₁/2 ⇒ a =2S /h₁ аналогично: b =2S/h₂ ; c = 2S/h₃. p =S(1/h₁+1/h₂+1/h₃) p -a = S(1/h₁+1/h₂+1/h₃) -2S/h₁ ) = S(1/h₂+1/h₃ -1/h₁) аналогично: p -b = S(1/h₁+1/h₂+1/h₃) -2S/h₂ ) = S(1/h₁+1/h₃ -1/h₂) p -c = S(1/h₁+1/h₂+1/h₃) -2S/h₃ ) = S(1/h₁+1/h₂ -1/h₃) . S =S²√((1/h₁+1/h₂+1/h₃)*(1/h₂+1/h₃ -1/h₁)(1/h₂+1/h₃ -1/h₁) *(1/h₁+1/h₂ -1/h₃) ) S =1:√((1/h₁+1/h₂+1/h₃)(1/h₂+1/h₃-1/h₁)(1/h₁+1/h₃-1/h₂)(1/h₁+1/h₂-1/h₃)) * * * 1/h₁+1/h₂+1/h₃=1/24 + 1/30 +1/40 = (5+4+3)/120 = 112/120 =1/10 * * * S =1: √((1/10)*(1/60)*(1/30)*(1/20))= 1: √((1/600)*(1/600))=600 (см²) . ответ : 600 см².
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Task/24841973---.---.---.---.---
знайти площу трикутника якщо його висоти дорівнюють 24, 30, 40 см
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По формуле Герона S =√p(p-a)(p-b)(p-c), где p =(a+b+c)/2_полупериметр
обозначаем h(a)=h₁ ; h(b)=h₂ ; h(c) =h₃
S =a*h₁/2 ⇒ a =2S /h₁
аналогично: b =2S/h₂ ; c = 2S/h₃.
p =S(1/h₁+1/h₂+1/h₃)
p -a = S(1/h₁+1/h₂+1/h₃) -2S/h₁ ) = S(1/h₂+1/h₃ -1/h₁)
аналогично:
p -b = S(1/h₁+1/h₂+1/h₃) -2S/h₂ ) = S(1/h₁+1/h₃ -1/h₂)
p -c = S(1/h₁+1/h₂+1/h₃) -2S/h₃ ) = S(1/h₁+1/h₂ -1/h₃) .
S =S²√((1/h₁+1/h₂+1/h₃)*(1/h₂+1/h₃ -1/h₁)(1/h₂+1/h₃ -1/h₁) *(1/h₁+1/h₂ -1/h₃) )
S =1:√((1/h₁+1/h₂+1/h₃)(1/h₂+1/h₃-1/h₁)(1/h₁+1/h₃-1/h₂)(1/h₁+1/h₂-1/h₃) )
* * * 1/h₁+1/h₂+1/h₃=1/24 + 1/30 +1/40 = (5+4+3)/120 = 112/120 =1/10 * * *
S =1: √((1/10)*(1/60)*(1/30)*(1/20))= 1: √((1/600)*(1/600))=600 (см²) .
ответ : 600 см².