Question

1. a(-6,-6)b(12,0) c(-3,7)d(3,-11) \overline{ab} and \overline{cd} are... m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} d = \sqrt{(y_{2}-y_{1})^{2}+(x_{2}-x_{1})^{2}} parallel and congruent parallel and not congruent perpendicular and congruent perpendicular and not congruent

Explanation:

Step1: Calculate the slope of \( \overline{AB} \)

Given \( A(-6,-6) \) and \( B(12,0) \), use the slope formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \).
\( m_{AB}=\frac{0 - (-6)}{12 - (-6)}=\frac{6}{18}=\frac{1}{3} \)

Step2: Calculate the slope of \( \overline{CD} \)

Given \( C(-3,7) \) and \( D(3,-11) \), use the slope formula.
\( m_{CD}=\frac{-11 - 7}{3 - (-3)}=\frac{-18}{6}=- 3 \)

Step3: Check if slopes are parallel (\( m_1=m_2 \)) or perpendicular (\( m_1\times m_2=-1 \))

\( m_{AB}\times m_{CD}=\frac{1}{3}\times(-3)=-1 \), so \( \overline{AB} \) and \( \overline{CD} \) are perpendicular.

Step4: Calculate the length of \( \overline{AB} \)

Use the distance formula \( d=\sqrt{(y_2 - y_1)^2+(x_2 - x_1)^2} \).
\( d_{AB}=\sqrt{(0 + 6)^2+(12 + 6)^2}=\sqrt{36 + 324}=\sqrt{360}=6\sqrt{10} \)

Step5: Calculate the length of \( \overline{CD} \)

\( d_{CD}=\sqrt{(-11 - 7)^2+(3 + 3)^2}=\sqrt{(-18)^2+6^2}=\sqrt{324 + 36}=\sqrt{360}=6\sqrt{10} \)

Answer:

perpendicular and congruent