Question

note that \\( \triangle l m n \\) has vertices \\( l(-3,6), m(-8,3) \\), and \\( n(-1,1) \\).
also, note that \\( \triangle l \prime m \prime n \prime \\) has vertices \\( l \prime(2,-2), m \prime(-3,-5) \\), and \\( n \prime(4,-7) \\).
complete the following.
(a) find each slope. give exact answers (not decimal approximations).
slope of \\( \overline{l l \prime}= \\)
slope of \\( \overline{m m \prime}= \\)
slope of \\( \overline{n n \prime}= \\)
(b) find each length. give exact answers (not decimal approximations).
\\( l l \prime= \\) units
\\( m m \prime= \\) units
\\( n n \prime= \\) units
(c) choose the correct statement about the translation.
all points on the original figure moved the same distance and in the same
direction.

Explanation:

Step1: Calculate slope of \( \overline{LL'} \)

The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For \( L(-3,6) \) and \( L'(2,-2) \), \( x_1=-3,y_1 = 6,x_2=2,y_2=-2 \).
\( m_{LL'}=\frac{-2 - 6}{2-(-3)}=\frac{-8}{5}\)

Step2: Calculate slope of \( \overline{MM'} \)

For \( M(-8,3) \) and \( M'(-3,-5) \), \( x_1=-8,y_1 = 3,x_2=-3,y_2=-5 \).
\( m_{MM'}=\frac{-5 - 3}{-3-(-8)}=\frac{-8}{5}\)

Step3: Calculate slope of \( \overline{NN'} \)

For \( N(-1,1) \) and \( N'(4,-7) \), \( x_1=-1,y_1 = 1,x_2=4,y_2=-7 \).
\( m_{NN'}=\frac{-7 - 1}{4-(-1)}=\frac{-8}{5}\)

Step4: Calculate length of \( \overline{LL'} \)

The distance formula is \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \). For \( L(-3,6) \) and \( L'(2,-2) \), \( d_{LL'}=\sqrt{(2 - (-3))^2+(-2 - 6)^2}=\sqrt{25 + 64}=\sqrt{89}\)

Step5: Calculate length of \( \overline{MM'} \)

For \( M(-8,3) \) and \( M'(-3,-5) \), \( d_{MM'}=\sqrt{(-3-(-8))^2+(-5 - 3)^2}=\sqrt{25+64}=\sqrt{89}\)

Step6: Calculate length of \( \overline{NN'} \)

For \( N(-1,1) \) and \( N'(4,-7) \), \( d_{NN'}=\sqrt{(4-(-1))^2+(-7 - 1)^2}=\sqrt{25 + 64}=\sqrt{89}\)

Answer:

(a) Slope of \( \overline{LL'}=-\frac{8}{5}\), Slope of \( \overline{MM'}=-\frac{8}{5}\), Slope of \( \overline{NN'}=-\frac{8}{5}\)
(b) \( LL'=\sqrt{89}\) units, \( MM'=\sqrt{89}\) units, \( NN'=\sqrt{89}\) units
(c) All points on the original figure moved the same distance and in the same direction.