Matemaatika:Gümnaasium/Kordamine eksamiks/Geomeetria tasandil

Valemid

Ruut

{\begin{matrix}d=a{\sqrt {2}}\end{matrix}}

{\begin{matrix}S=a^{2}\end{matrix}}

Kolmnurk

S={\frac {b*h}{2}}={\frac {a*b*sin\gamma }{2}}

S={\sqrt {p(p-a)(p-b)(p-c)}},

kus

p={\frac {1}{2}}(a+b+c)

\left.{\begin{matrix}a^{2}=b^{2}+c^{2}-2bc*cos\alpha \\b^{2}=a^{2}+c^{2}-2ac*cos\beta \\c^{2}=a^{2}+b^{2}-2ab*cos\gamma \end{matrix}}\right\}

Koosinusteoreem

{\frac {a}{sin\alpha }}={\frac {b}{sin\beta }}={\frac {c}{sin\gamma }}=2R

Siinusteoreem

{\begin{matrix}\alpha +\beta +\gamma =180^{0}\\{\mbox{R - }}{\ddot {u}}{\mbox{mberringjoone raadius}}\end{matrix}}

Täisnurkne kolmnurk

$a^{2}+b^{2}=c^{2}\Rightarrow c={\sqrt {a^{2}+b^{2}}}$ Pythagorase teoreem

$S={\frac {ab}{2}}={\frac {ch}{2}}$

$sin\alpha ={\frac {a}{c}}=cos\beta$

$cos\alpha ={\frac {b}{c}}=sin\beta$

$tan\alpha ={\frac {a}{b}}$

${\begin{matrix}\alpha +\beta =90^{0}\\a^{2}=f*c;&sin(90^{0}-\alpha )=cos\alpha \\b^{2}=g*c;&cos(90^{0}-\alpha )=sin\alpha \\h^{2}=f*g;&tan(90^{0}-\alpha )={\frac {1}{tan\alpha }}\end{matrix}}$

Ringjoon, ring, sektor

Ring

{\begin{matrix}C=2*\pi *r=\pi *d\end{matrix}}

, C on ümberringjoone pikkus

S=\pi *r^{2}={\frac {\pi *d^{2}}{4}}

, S on täispindala

S_{s}={\frac {\pi *r^{2}}{2\pi }}*x={\frac {x*r^{2}}{2}}={\frac {l*r}{2}}

, S_s on sektori pinda

l={\frac {2*\pi *r}{2\pi }}*x=x*r

, l on sektori kaare pikkus

Ristkülik

{\begin{matrix}d={\sqrt {a^{2}+b^{2}}}\end{matrix}}

{\begin{matrix}S=a*b\end{matrix}}

{\begin{matrix}P=2(a+b)\end{matrix}}

Romb

{\begin{matrix}S=a*h\end{matrix}}

{\begin{matrix}S=a^{2}sin\alpha \end{matrix}}

{\begin{matrix}S={\frac {1}{2}}d_{1}d_{2}\end{matrix}}

Näited

Rombi ümber asetseb minimaalse suurusega ring. Leia mitu korda on romb ringist väiksem, kui antud on rombi lühem diagonaal ja alus.

$h={\sqrt {d_{2}^{2}-{\frac {a^{2}}{4}}}}$

$S_{romb}=a*{\sqrt {d_{2}^{2}-{\frac {a^{2}}{4}}}}$

$d_{1}={\frac {a*{\sqrt {d_{2}^{2}-{\frac {a^{2}}{4}}}}}{d_{2}}}$

$r={\frac {\frac {a*{\sqrt {d_{2}^{2}-{\frac {a^{2}}{4}}}}}{d_{2}}}{2}}$

$S_{ring}=\pi *{\frac {\frac {a*{\sqrt {d_{2}^{2}-{\frac {a^{2}}{4}}}}}{d_{2}}}{2}}^{2}$

$x={\frac {S_{ring}}{S_{romb}}}\Rightarrow x={\frac {\pi *{\frac {\frac {a*{\sqrt {d_{2}^{2}-{\frac {a^{2}}{4}}}}}{d_{2}}}{2}}^{2}}{a*{\sqrt {d_{2}^{2}-{\frac {a^{2}}{4}}}}}}\Rightarrow x=\pi *{\frac {4a^{2}*{\frac {d_{2}^{2}-{\frac {a^{2}}{4}}}{d_{2}^{2}}}}{a*{\sqrt {d_{2}^{2}-{\frac {a^{2}}{4}}}}}}\Rightarrow$

$\Rightarrow x=4\pi a{\frac {(d_{2}^{2}-{\frac {a^{2}}{4}})*{\sqrt {d_{2}^{2}-{\frac {a^{2}}{4}}}}}{d_{2}^{2}}}\Rightarrow$

$\Rightarrow x=(4\pi a-{\frac {\pi a^{3}}{d_{2}^{2}}}){\sqrt {d_{2}^{2}+{\frac {a^{2}}{4}}}}$

Vastus: Romb on ringist $(4\pi a-{\frac {\pi a^{3}}{d_{2}^{2}}}){\sqrt {d_{2}^{2}+{\frac {a^{2}}{4}}}}$ korda väiksem.