Question

24 multiple choice 2 points a counterexample to the following statement. \lines are parallel if they do not intersect\ is: there are no counter examples two line segments that intersect the two longer sides of a rectangle two lines on different sheets of paper 25 multiple choice 2 points which equation represents a line that is perpendicular to the graph of the line -2y = 4x + 10? y = \frac{1}{2}x + 6 y = -2x + 6 y = -\frac{1}{2}x + 6 y = 2x + 6

Explanation:

Question 24

• Option 1: Incorrect. There are counter - examples.
• Option 2: Incorrect. The statement is about non - intersecting lines, and this option is about intersecting line segments.
• Option 3: Incorrect. The two longer sides of a rectangle are parallel (they don't intersect and are in the same plane).
• Option 4: Correct. Two lines on different sheets of paper (in different planes) that do not intersect are not parallel. Parallel lines must be in the same plane.

Step1: Find the slope of the given line

The equation of the given line is \(-2y = 4x+10\). Solve for \(y\) to get it in slope - intercept form \(y = mx + b\) (where \(m\) is the slope).
Divide both sides of \(-2y=4x + 10\) by \(-2\):
\(y=-2x - 5\). So the slope of the given line \(m_1=-2\).

Step2: Find the slope of the perpendicular line

If two lines with slopes \(m_1\) and \(m_2\) are perpendicular, then \(m_1\times m_2=-1\). Let \(m_1=-2\), then \(-2\times m_2=-1\). Solve for \(m_2\):
\(m_2=\frac{-1}{-2}=\frac{1}{2}\)

Step3: Check the slopes of the given options

• For \(y = \frac{1}{2}x+6\), the slope \(m=\frac{1}{2}\)
• For \(y=-2x + 6\), the slope \(m=-2\)
• For \(y=-\frac{1}{2}x+6\), the slope \(m=-\frac{1}{2}\)
• For \(y = 2x+6\), the slope \(m = 2\)

Answer:

two lines on different sheets of paper

Question 25