Question

teks a.5.c
lesson
5 - 8
practice b
slopes of parallel and perpendicular lines
identify which lines are parallel.

1. ( y = 3x + 4 ); ( y = 4 ); ( y = 3x ); ( y = 3 )
2. ( y=\frac{1}{2}x + 4 ); ( x=\frac{1}{2} ); ( 2x + y = 1 ); ( y=\frac{1}{2}x + 1 )
3. find the slope of each segment.

slope of ( overline{ab} ):
slope of ( overline{ad} ):
slope of ( overline{dc} ):
slope of ( overline{bc} ):
explain why ( abcd ) is a parallelogram.

identify which lines are perpendicular.

1. ( y = 5 ); ( y=\frac{1}{8}x ); ( x = 2 ); ( y = 8x - 5 )
2. ( y = - 2 ); ( y=-\frac{1}{2}x - 4 ); ( y - 4 = 2(x + 3) ); ( y = - 2x )
3. show that ( abc ) is a right triangle.

Explanation:

Step1: Find slope of \(AB\)

The formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For \(A(-4,3)\) and \(B(2,-1)\), \(m_{AB}=\frac{-1 - 3}{2-(-4)}=\frac{-4}{6}=-\frac{2}{3}\)

Step2: Find slope of \(AD\)

For \(A(-4,3)\) and \(D(-4,-1)\), \(m_{AD}=\frac{-1 - 3}{-4-(-4)}=\frac{-4}{0}\), slope is undefined (vertical line)

Step3: Find slope of \(DC\)

For \(D(-4,-1)\) and \(C(2,-5)\), \(m_{DC}=\frac{-5 - (-1)}{2-(-4)}=\frac{-4}{6}=-\frac{2}{3}\)

Step4: Find slope of \(BC\)

For \(B(2,-1)\) and \(C(2,-5)\), \(m_{BC}=\frac{-5 - (-1)}{2 - 2}=\frac{-4}{0}\), slope is undefined (vertical line)

Step5: Explain parallelogram

A parallelogram has opposite sides parallel. \(m_{AB}=m_{DC}=-\frac{2}{3}\), so \(AB\parallel DC\). \(m_{AD}\) and \(m_{BC}\) are both undefined (vertical lines), so \(AD\parallel BC\)

Answer:

slope of \(\overline{AB}\): \(-\frac{2}{3}\)
slope of \(\overline{AD}\): undefined
slope of \(\overline{DC}\): \(-\frac{2}{3}\)
slope of \(\overline{BC}\): undefined
\(ABCD\) is a parallelogram because \(AB\parallel DC\) (equal slopes \( -\frac{2}{3}\)) and \(AD\parallel BC\) (both vertical lines, undefined slope)