Additional Practice Problems: Answers

\begin{document} \maketitle 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} \end{document}