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Zadatak 0009

Test: MPsIIIBGIII_001

 

Dat je trougao ABC tako da je \(A(0,1), B(-2, 5), C(4,9)\). Odrediti:

a) jednačinu stranice AB
b) jednačinu težišne duži tb
c) jednačinu visine hc
d) ugao α
e) površinu trougla ABC

 

Rešenje:


 

a)\(A(0,1), B(-2, 5)\Rightarrow 1=k\cdot 0+n\wedge 5=k\cdot (-2)+n\)

 

\(\begin{array}{rcl}n & = & 1 \\ -2k+1 & = & 5\end{array}\)

 

\(\begin{array}{rcl}n & = & 1 \\ k & = & -2 \end{array}\)

 

\(y=-2x+1\).

 

b) Težišna duž tb određena je tačkama B i B1, gde je B1 sredeište duži AC.

 

\( x_{B_{1}}=\frac{0+4}{2}=2, y_{B_{1}}=\frac{1+9}{2}=5\Rightarrow B_{1}=(2,5)\)

 

\(B(-2,5), B_{1}(2,5)\Rightarrow 5=k\cdot (-2)+n\wedge 5=k\cdot 2+n\)

 

\(\begin{array}{rcl}-2k+n & = & 5 \\ 2k+n & = & 5 \end{array}\)

 

\(\begin{array}{rcl}n & = & 5 \\ k & = & 0 \end{array}\)

 

\(\Rightarrow y=5\).

 

c) Potrebno je odrediti pravu koja sadrži tačku C i upravna je na pravu određenu tačkama A i B.

 

\(AB:y=-2x+1\Rightarrow k_{h_{c}}=\frac{1}{2}\)

 

\(9=\frac{1}{2}\cdot 4+n\Rightarrow n=7\)

 

\(h_{c}:y=\frac{1}{2}x+7\).

 

d) \(\alpha =\measuredangle \left ( AB, AC \right )\)

 

\(A(0,1), C(4, 9)\Rightarrow 1=k\cdot 0+n\wedge 9=k\cdot 4+n\) 

 

\(\begin{array}{rcl}n & = & 1 \\ 4k+1 & = & 9\end{array}\)

 

\(\begin{array}{rcl}n & = & 1 \\ k & = & 2 \end{array}\)

 

\(AB:y=-2x+1\)

 

\(AC:y=2x+1\)

 

\( tg\varphi =\frac{k_{2}-k_{1}}{1+k_{1}k_{2}}=\frac{4}{1-4}=-\frac{4}{3}\)

 

\(\varphi =\arctan\left (-\frac{4}{3}  \right )\).

 

e) \(P= \frac{1}{2}\mid \begin{vmatrix}
0 & 1 &1 \\
-2 & 5 & 1\\
4 & 9  & 1
\end{vmatrix}\mid =\frac{1}{2}\cdot \mid 4-18-20+2\mid =16\).