Consider the graph of the line [tex]y=\frac{1}{2} x-4[/tex] and the point $(-4,2)$.
The slope of a line parallel to the given line is $\square$.
A point on the line parallel to the given line, passing through ( $-4,2$ ), is $\square$.
The slope of a line perpendicular to the given line is $\square$
A point on the line perpendicular to the given line, passing through $(-4,2)$, is $\square$
Answer (2)
The slope of a line parallel to y = 2 1 x − 4 is the same as the slope of the given line: 2 1 .
Find the equation of the parallel line passing through ( − 4 , 2 ) : y = 2 1 x + 4 , and identify a point on this line: ( 0 , 4 ) .
Determine the slope of a line perpendicular to the given line by taking the negative reciprocal of the original slope: − 2 .
Find the equation of the perpendicular line passing through ( − 4 , 2 ) : y = − 2 x − 6 , and identify a point on this line: ( 0 , − 6 ) .
The slope of the parallel line is 2 1 , a point on it is ( 0 , 4 ) , the slope of the perpendicular line is − 2 , and a point on it is ( 0 , − 6 ) .
Explanation
Problem Analysis The given line is y = 2 1 x − 4 . We are also given a point ( − 4 , 2 ) . We need to find the slope of a line parallel to the given line, a point on the line parallel to the given line and passing through the given point, the slope of a line perpendicular to the given line, and a point on the line perpendicular to the given line and passing through the given point.
Slope of Parallel Line The slope of a line parallel to the given line is the same as the slope of the given line. The slope of the given line y = 2 1 x − 4 is 2 1 .
Point on Parallel Line The equation of the line parallel to the given line and passing through ( − 4 , 2 ) is y = 2 1 x + b . Substituting ( − 4 , 2 ) into the equation to find b : 2 = 2 1 ( − 4 ) + b , so 2 = − 2 + b , which means b = 4 . The equation of the parallel line is y = 2 1 x + 4 . To find a point on this line, we can choose any x value and calculate the corresponding y value. Let x = 0 , then y = 2 1 ( 0 ) + 4 = 4 . So, the point ( 0 , 4 ) is on the line.
Slope of Perpendicular Line The slope of a line perpendicular to the given line is the negative reciprocal of the slope of the given line. The slope of the given line is 2 1 , so the slope of the perpendicular line is − 2 .
Point on Perpendicular Line The equation of the line perpendicular to the given line and passing through ( − 4 , 2 ) is y = − 2 x + c . Substituting ( − 4 , 2 ) into the equation to find c : 2 = − 2 ( − 4 ) + c , so 2 = 8 + c , which means c = − 6 . The equation of the perpendicular line is y = − 2 x − 6 . To find a point on this line, we can choose any x value and calculate the corresponding y value. Let x = 0 , then y = − 2 ( 0 ) − 6 = − 6 . So, the point ( 0 , − 6 ) is on the line.
Final Answer The slope of a line parallel to the given line is 2 1 . A point on the line parallel to the given line, passing through ( − 4 , 2 ) , is ( 0 , 4 ) . The slope of a line perpendicular to the given line is − 2 . A point on the line perpendicular to the given line, passing through ( − 4 , 2 ) , is ( 0 , − 6 ) .
Examples
Understanding parallel and perpendicular lines is crucial in architecture and design. For instance, when designing a building, architects use parallel lines for walls and perpendicular lines to ensure structural stability at corners. Knowing how to find these lines helps in creating precise and safe structures. Imagine designing a rectangular room; the walls are parallel, and the corners where they meet are perpendicular, ensuring the room's integrity and aesthetic appeal.
The slope of a line parallel to the given line is 2 1 , and a point on it is ( 0 , 4 ) . The slope of a line perpendicular to the given line is − 2 , and a point on it is ( 0 , − 6 ) .
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