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vodka [1.7K]
3 years ago
7

The coordinates of the vertices of triangle ABC are A(-1,3), B(1,2) and C(-3,-1).

Mathematics
1 answer:
Vlad1618 [11]3 years ago
5 0
Check the picture below.

so... let's check those slopes then.  Notice the triangle, the triangle "seem" to have a right-angle at the vertex A, namely, the slopes of AB and AC are perpendicular if that's true.

Now, when the product of two perpendicular slopes is -1.

so, let's check the slopes then.

\bf \begin{array}{ccccccccc}
&&x_1&&y_1&&x_2&&y_2\\
%  (a,b)
&A&(~{{ -1}} &,&{{ 3}}~) 
%  (c,d)
&B&(~{{ 1}} &,&{{ 2}}~)
\end{array}
\\\\\\
% slope  = m
slope = {{ m}}\implies 
\cfrac{\stackrel{rise}{{{ y_2}}-{{ y_1}}}}{\stackrel{run}{{{ x_2}}-{{ x_1}}}}\implies \cfrac{2-3}{1-(-1)}\implies \cfrac{2-3}{1+1}\implies \cfrac{-1}{2}\\\\
-------------------------------\\\\

\bf \begin{array}{ccccccccc}
&&x_1&&y_1&&x_2&&y_2\\
%  (a,b)
&A&(~{{ -1}} &,&{{ 3}}~) 
%  (c,d)
&C&(~{{ -3}} &,&{{ -1}}~)
\end{array}
\\\\\\
% slope  = m
slope = {{ m}}\implies 
\cfrac{\stackrel{rise}{{{ y_2}}-{{ y_1}}}}{\stackrel{run}{{{ x_2}}-{{ x_1}}}}\implies \cfrac{-1-3}{-3-(-1)}\implies \cfrac{-1-3}{-3+1}
\\\\\\
\cfrac{-4}{-2}\implies \cfrac{2}{1}\implies 2\\\\
-------------------------------

\bf \begin{array}{ccccccccc}
&&x_1&&y_1&&x_2&&y_2\\
%  (a,b)
&C&(~{{ -3}} &,&{{ -1}}~) 
%  (c,d)
&B&(~{{ 1}} &,&{{ 2}}~)
\end{array}
\\\\\\
% slope  = m
slope = {{ m}}\implies 
\cfrac{\stackrel{rise}{{{ y_2}}-{{ y_1}}}}{\stackrel{run}{{{ x_2}}-{{ x_1}}}}\implies \cfrac{2-(-1)}{1-(-3)}\implies \cfrac{2+1}{1+3}\implies \cfrac{3}{4}

now, let's check the product of AB and AC,   \bf \stackrel{AB_m}{-\cfrac{1}{2}}\cdot \stackrel{AC_m}{2}\implies -1

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