Question

geometry a
unit 1 (m1t1l1 - 4) test c
learning target: i can identify and write the equations of lines perpendicular to given lines

1. are the following 2 lines with the slopes below parallel, perpendicular, or neither? justify

your thinking.
a) ( m_1=-4 ) and ( m_2 = 4 )
b) ( m_1=\frac{8}{5} ) and ( m_2=-\frac{5}{8} )
c) ( m_1=\frac{2}{3} ) and ( m_2=\frac{2}{3} )

1. write the equation of a line parallel to ( y=-3x - 7 ) through the point ( (-2,-4) ). put your

answer in slope intercept form ( (y = mx + b) ).

1. write the equation of a line perpendicular to ( y=\frac{7}{4}x + 11 ) through the point ( (-28,9) ). put

your answer in slope intercept form ( (y = mx + b) ).

Explanation:

Step1: Find the slope of the parallel line

For a line parallel to \(y = -3x - 7\), the slope \(m\) is the same as the slope of the given line. So \(m=-3\).

Step2: Use the point - slope form \(y - y_1=m(x - x_1)\)

We have \(m = - 3\), \(x_1=-2\), \(y_1=-4\). Substitute into the formula:
\(y-(-4)=-3(x - (-2))\)
\(y + 4=-3(x + 2)\)

Step3: Convert to slope - intercept form \(y=mx + b\)

Expand the right - hand side: \(y + 4=-3x-6\)
Subtract 4 from both sides: \(y=-3x-6 - 4\)
\(y=-3x-10\)

Step4: Find the slope of the perpendicular line

For a line perpendicular to \(y=\frac{7}{4}x + 11\), the slope \(m\) is the negative reciprocal of \(\frac{7}{4}\). So \(m=-\frac{4}{7}\)

Step5: Use the point - slope form \(y - y_1=m(x - x_1)\)

We have \(m =-\frac{4}{7}\), \(x_1=-28\), \(y_1 = 9\). Substitute into the formula:
\(y - 9=-\frac{4}{7}(x-(-28))\)
\(y - 9=-\frac{4}{7}(x + 28)\)

Step6: Convert to slope - intercept form \(y=mx + b\)

Expand the right - hand side: \(y-9=-\frac{4}{7}x-16\)
Add 9 to both sides: \(y=-\frac{4}{7}x-16 + 9\)
\(y=-\frac{4}{7}x-7\)

Answer:

1.

• a) Neither. Since \(m_1=-4\), \(m_2 = 4\), \(m_1

eq m_2\) (not parallel) and \(m_1\times m_2=-16
eq - 1\) (not perpendicular)

• b) Perpendicular. Since \(m_1=\frac{8}{5}\), \(m_2=-\frac{5}{8}\), \(m_1\times m_2=\frac{8}{5}\times(-\frac{5}{8})=-1\)
• c) Parallel. Since \(m_1=\frac{2}{3}\), \(m_2=\frac{2}{3}\), \(m_1 = m_2\)

1. \(y=-3x-10\)
2. \(y=-\frac{4}{7}x-7\)