Line EF has an equation of [tex]$y=-2x+7$[/tex]. Which of the following could be an equation for a line that is perpendicular to line EF?
A. [tex]$y=2x-3$[/tex]
B. [tex]$y=\frac{1}{2}x-3$[/tex]
C. [tex]$y=-2x-3$[/tex]
D. [tex]$y=-\frac{1}{2}x-3$[/tex]
Answer (2)
Identify the slope of the given line EF as − 2 .
Calculate the negative reciprocal of the slope, which is 2 1 .
Check the given options for a line with a slope of 2 1 .
The equation of the perpendicular line is y = 2 1 x − 3 .
Explanation
Understanding the Problem The problem asks us to find the equation of a line that is perpendicular to a given line. The equation of the given line EF is y = − 2 x + 7 . We need to identify which of the provided options represents a line perpendicular to EF.
Key Concept: Perpendicular Lines The key concept here is that perpendicular lines have slopes that are negative reciprocals of each other. This means if one line has a slope of m , a line perpendicular to it will have a slope of − m 1 .
Finding the Slope of Line EF First, let's identify the slope of the given line EF. The equation y = − 2 x + 7 is in slope-intercept form ( y = m x + b ), where m represents the slope. In this case, the slope of line EF is − 2 .
Calculating the Negative Reciprocal Now, we need to find the negative reciprocal of − 2 . The negative reciprocal is calculated as follows: − − 2 1 = 2 1 . So, any line perpendicular to line EF must have a slope of 2 1 .
Identifying the Perpendicular Line We now examine the given options to find the line with a slope of 2 1 :
y = 2 x − 3 (slope is 2)
y = 2 1 x − 3 (slope is 2 1 )
y = − 2 x − 3 (slope is -2)
y = − 2 1 x − 3 (slope is − 2 1 )
Only the equation y = 2 1 x − 3 has the required slope of 2 1 .
Final Answer Therefore, the equation of a line perpendicular to line EF is y = 2 1 x − 3 .
Examples
Understanding perpendicular lines is crucial in architecture and construction. When designing buildings, ensuring walls are perpendicular to the ground or that different sections of a structure meet at right angles is essential for stability and safety. The concept of negative reciprocal slopes helps architects calculate these angles accurately, ensuring structural integrity.
The equation of a line that is perpendicular to line EF, given by y = − 2 x + 7 , is y = 2 1 x − 3 . This is determined by finding the negative reciprocal of the slope of line EF. Therefore, the correct option is B.
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