Use the given line and the point not on the line to answer the question. What is the point on the line perpendicular to the given line, passing through the given point that is also on the $y$-axis?
$(-3.6,0)$
$(-2,0)$
$(0,-3.6)$
$(0,-2)$
Answer (2)
Calculate the slope of the line through ( − 3.6 , 0 ) and ( − 2 , 0 ) , which is m 1 = 0 .
Determine the slope of the perpendicular line, which is undefined, indicating a vertical line.
Recognize that the perpendicular line passing through either ( 0 , − 3.6 ) or ( 0 , − 2 ) is the y-axis, x = 0 .
Conclude that the point on the y-axis is the given point itself: ( 0 , − 2 ) (assuming the point is ( 0 , − 2 ) ).
Explanation
Problem Analysis We are given two points on a line: ( − 3.6 , 0 ) and ( − 2 , 0 ) . We need to find a point on the line perpendicular to this line that passes through a given point and lies on the y-axis. The possible given points are ( 0 , − 3.6 ) and ( 0 , − 2 ) .
Find the slope of the given line First, let's find the slope of the line passing through ( − 3.6 , 0 ) and ( − 2 , 0 ) . The slope, m 1 , is calculated as: m 1 = x 2 − x 1 y 2 − y 1 = − 2 − ( − 3.6 ) 0 − 0 = 1.6 0 = 0 Since the slope is 0, the line is a horizontal line (y = 0).
Find the slope of the perpendicular line Now, we need to find the slope of a line perpendicular to the given line. If the original line has a slope of m 1 , the perpendicular line has a slope of m 2 such that m 1 m 2 = − 1 . Since m 1 = 0 , a line perpendicular to it must be a vertical line. Vertical lines have an undefined slope, and their equation is of the form x = c , where c is a constant.
Find the equation of the perpendicular line We are given two possible points through which the perpendicular line passes: ( 0 , − 3.6 ) and ( 0 , − 2 ) . Since the perpendicular line is vertical, its equation must be x = c . If the line passes through ( 0 , − 3.6 ) , the equation is x = 0 . If the line passes through ( 0 , − 2 ) , the equation is x = 0 . In both cases, the perpendicular line is the y-axis itself.
Find the point on the y-axis We are looking for the point on the line perpendicular to the given line that is also on the y-axis. Since the perpendicular line is the y-axis (x = 0), and we are looking for a point on the y-axis, the x-coordinate of the point must be 0. The question states that the point is on the y-axis. Therefore, the point we are looking for is one of the given points ( 0 , − 3.6 ) or ( 0 , − 2 ) . However, the question is ambiguous as it asks for the point, implying a unique solution. Since the perpendicular line must pass through the given point, the answer must be either ( 0 , − 3.6 ) or ( 0 , − 2 ) . Without further information, we cannot determine which point is the correct answer. However, since the question asks for a point on the line perpendicular to the given line, passing through the given point, the answer must be the point itself. Let's assume the point is ( 0 , − 2 ) . Then the perpendicular line is x = 0 , and the point on this line that is also on the y-axis is ( 0 , − 2 ) .
Final Answer Therefore, assuming the point through which the perpendicular line passes is ( 0 , − 2 ) , the point on the line perpendicular to the given line that is also on the y-axis is ( 0 , − 2 ) .
Examples
Understanding perpendicular lines is crucial in architecture and construction. When designing a building, ensuring walls are perpendicular to the ground (horizontal plane) is essential for stability. Similarly, in road construction, perpendicular intersections are carefully planned to ensure safe and efficient traffic flow. This concept is also used in creating accurate maps and navigation systems, where perpendicular lines help define coordinates and directions.
The point on the line perpendicular to the given horizontal line, which passes through a point on the y-axis, is ( 0 , − 2 ) . This is chosen as it is assumed to be the relevant solution in the absence of additional context. Other possible points could also exist but were not specified.
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