Esercizio 12.11 Trovare una retta \( r \) passante per il punto \( P=(1,-1,0) \) e parallela al piano \( \pi: 2 x-y+4=0 \). E unica tale retta? Svolgimento Le rette cercate sono infinite, sono tutte quelle che giacciono nel piano parallelo a \( \pi \) e passante per \( P \), che passano a loro volta per \( P \). Il piano che le contiene è \( \pi^{\prime}: 2 x-y=3 \). Per trovare una retta con quella proprieta, basta intersecare \( \pi^{\prime} \) con un piano passante per \( P \). Ad esempio \( \sigma: z=0 \). Allora si ottiene \( r:\left\{\begin{array}{l}2 x-y=3 \\ z=0 .\end{array}\right. \)

13. Una lámpara está instalada en lo alto de un poste de 15 pies de altura. Un hombre de 6 pies de estatura se aleja caminando desde el poste con una rapidez de 5 pies/s a lo largo de una trayectoria rectilínea. ¿Qué tan rápido se desplaza la punta de su sombra cuando el hombre está a 40 pies del poste?

A ball bounces \( \frac{1}{2} \) of its previous height on each bounce. Find the total distance covered by the ball, until it comes to rest, if it is dropped from a height of 20 metres.

\begin{tabular}{l|l|}\hline \( \begin{array}{l}\text { Sample Response: Because vertical angles are } \\ \text { congruent, angles } 1 \text { and } 4 \text { are congruent. Because } \\ \text { corresponding angles are congruent, angles } 4 \text { and } 8 \text { are } \\ \begin{array}{l}\text { congruent. Because angle } 4 \text { is congruent to both } 1 \text { and } 8, \\ \text { angle } 1 \text { is congruent to angle } 8 .\end{array}\end{array} \) & \( \begin{array}{l}\text { What did you include in your response? Check all that } \\ \text { apply. } \\ \square \text { vertical angles theorem } \\ \square \text { corresponding angles theorem } \\ \square \text { transitive property }\end{array} \) \\ \( \square \) supplementary angles \\ \( \square \) linear pairs \end{tabular}

Determine if the following lines are parallel, perpendicular, or neither \[ \begin{array}{l}y=\frac{8 x}{7}-1 \\ -x+3 y=1\end{array} \]

Find the perpendicular d \( x^{2}+y^{2}-8 x+2 y+8 \) Hence, find the shortest