Question

name
2-4 reteach to build understanding
slopes of parallel and perpendicular lines
envision geometry
savvasrealize.com

1. match each set of lines with the correct statement.

(graphs of lines a, b in coordinate plane)
line a is parallel to line b.
( a parallel b )
the slope of line a times the slope of line b equals −1.
(graphs of lines a, b in coordinate plane)
line a is perpendicular to line b.
( a perp b )
the slope of line a equals the slope of line b.

1. fill in the blanks to find the equation of the line perpendicular to ( y = \frac{1}{3}x - 2 ) through the point (1, 4).

the slope of the given line is __. perpendicular lines have slopes with a product of , so the slope of the perpendicular line is . to write the equation of the perpendicular line, use the __ form to solve for the y-intercept.
( y = mx + b )
__ = −3 (__) + b
( b = ) ____
the equation of the line perpendicular to ( y = \frac{1}{3}x - 2 ) passing through the point (1, 4) is ____.

1. danielle says that the line perpendicular to ( y = 5x + 9 ) and passing through the point (3, 4) is ( y = 5x - 11 ). what is danielle’s error? how would you correct the error?

Explanation:

Problem 1:

For parallel lines (\(a \parallel b\)), their slopes are equal, so "Line \(a\) is parallel to line \(b\)" matches "The slope of line \(a\) equals the slope of line \(b\)". For perpendicular lines (\(a \perp b\)), the product of their slopes is \(-1\), so "Line \(a\) is perpendicular to line \(b\)" matches "The slope of line \(a\) times the slope of line \(b\) equals \(-1\)".

Step 1: Find slope of given line

The given line is \(y = \frac{1}{3}x - 2\), in slope - intercept form \(y=mx + b\) (where \(m\) is slope), so slope of given line \(m_1=\frac{1}{3}\).

Step 2: Recall perpendicular slope property

Perpendicular lines have slopes with a product of \(- 1\). Let slope of perpendicular line be \(m_2\), then \(m_1\times m_2=-1\). Substituting \(m_1 = \frac{1}{3}\), we get \(\frac{1}{3}\times m_2=-1\), so \(m_2=-3\).

Step 3: Use slope - intercept form

To write the equation of the perpendicular line, use the slope - intercept form \(y = mx + b\) to solve for the \(y\) - intercept. We know the point \((1,4)\) lies on the perpendicular line and \(m=-3\). Substitute \(x = 1\), \(y = 4\) and \(m=-3\) into \(y=mx + b\): \(4=-3(1)+b\).

Step 4: Solve for \(b\)

Simplify \(4=-3 + b\), add 3 to both sides: \(b=4 + 3=7\).

Step 5: Write the equation

The equation of the line with \(m=-3\) and \(b = 7\) is \(y=-3x + 7\).

1. Identify the error: The slope of a line perpendicular to \(y = 5x+9\) (slope \(m = 5\)) should be the negative reciprocal of \(5\), which is \(-\frac{1}{5}\), not \(5\) (Danielle used the same slope as the original line, which is for parallel lines, not perpendicular).
2. Correct the error:

• Step 1: Find the slope of the perpendicular line. For \(y = 5x + 9\) (slope \(m_1 = 5\)), the slope of the perpendicular line \(m_2=-\frac{1}{5}\) (since \(m_1\times m_2=-1\)).
• Step 2: Use the point - slope form \(y - y_1=m(x - x_1)\) with point \((3,4)\) and \(m=-\frac{1}{5}\).

\(y - 4=-\frac{1}{5}(x - 3)\)
\(y-4=-\frac{1}{5}x+\frac{3}{5}\)
\(y=-\frac{1}{5}x+\frac{3}{5}+4\)
\(y=-\frac{1}{5}x+\frac{3 + 20}{5}\)
\(y=-\frac{1}{5}x+\frac{23}{5}\) (or \(y=-0.2x + 4.6\)).

Answer:

• Line \(a\) is parallel to line \(b\) (\(a \parallel b\)): The slope of line \(a\) equals the slope of line \(b\).
• Line \(a\) is perpendicular to line \(b\) (\(a \perp b\)): The slope of line \(a\) times the slope of line \(b\) equals \(-1\).

Problem 2: