Additional Practice Problems: Answers   
20sep13


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Check your (written !) answer-hints to the practice problems. Be sure supply the details
of any steps you do not understand. Also, work through all indicated calculations.


\textbf{Problem 1}

\begin{itemize}
\item Substitute into $ X=aA+bB+cC $ what is given. Then 
$ X = 0.3A + bB + 0C $. Because $ a+b+c=1$ 
you can solve for $ b = 0.7 $. 
\end {itemize}

\textbf{Problem 2}

\begin{itemize}
\item There is an easy ways of solving this problem. Using Ceva's theorem, you can 
find the value of $ \frac{C'-A}{C'-B} $ and solve that for the barycentric
coordinates of $ C' $. Be sure you know how to sketch the corresponding figure
also freehand, not with some computer geometry tool. 
\item A second solution is to express $ (A A') $ and $ (B B') $ in parametric form
(be sure to use different letters for the parameter.) Then find $ G=(A A')(B B') $
by getting three equations in two unknowns (remember, barycentric coordinates are unique.)
They have a unique solution. Now repeat the process for $ C' = (AB)(CG) $.
\item By doing this problem both ways, you are checking your arithmetic, which is easy to make
mistakes in. Which method is less work? 
\end{itemize}

\textbf{Problems 3}

\begin{itemize}
\item
The point $ X = aA + bB + cC $ must have $ a = 0 $ and $ c=2b $.
Substituting and remembering that $ 1 = a+b+c $ you can solve the three equations for
a unique answer. Be sure you know how to sketch a believable figure for this. You can 
estimate the ratios by sight.
\end{itemize}

\textbf{Problems 4}

\begin{itemize}
\item 
Observe that from the hypothesis all four points are collinear. So you can put the real numberson the line and treat these differences as real numbers. Draw a figure for a guide.
\end{itemize}

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