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 defined by the points ( − 3.6 , 0 ) and ( − 2 , 0 ) , which is m 1 = 0 .
Determine the slope of the line perpendicular to the given line, which is m 2 = ∞ .
Find the equation of the perpendicular line passing through ( 0 , − 2 ) , which is x = 0 .
Identify the point on the y -axis where the perpendicular line intersects, which is ( 0 , − 2 ) .
Explanation
Problem Analysis We are given two points on a line: ( − 3.6 , 0 ) and ( − 2 , 0 ) . We are also given a point not on the line: ( 0 , − 2 ) . The objective is to find the point on the line perpendicular to the given line, passing through the given point that is also on the y -axis.
Calculate the slope of the given line First, we need to find the slope of the line defined by the points ( − 3.6 , 0 ) and ( − 2 , 0 ) . The slope, m 1 , is calculated as follows: m 1 = x 2 − x 1 y 2 − y 1 = − 2 − ( − 3.6 ) 0 − 0 = 1.6 0 = 0
Calculate the slope of the perpendicular line Next, we find the slope of the line perpendicular to the given line. Let this slope be m 2 . Since the lines are perpendicular, m 1 ⋅ m 2 = − 1 . In this case, since m 1 = 0 , the perpendicular line is vertical, and its slope is undefined (or infinite). We can represent this as: m 2 = m 1 − 1 = 0 − 1 = ∞
Find the equation of the perpendicular line Now, we write the equation of the line with slope m 2 passing through the point ( 0 , − 2 ) . Since the slope is infinite, the line is vertical and has the equation x = c , where c is a constant. Since the line passes through ( 0 , − 2 ) , the equation of the line is x = 0 .
Find the intersection point on the y-axis Finally, we need to find the point on the y -axis where the perpendicular line intersects. Since the equation of the perpendicular line is x = 0 , this is the y -axis. The line passes through the point ( 0 , − 2 ) , which is already on the y -axis. Therefore, the point on the line perpendicular to the given line, passing through the given point that is also on the y -axis is ( 0 , − 2 ) .
Final Answer The point on the line perpendicular to the given line, passing through the given point that is also on the y -axis is ( 0 , − 2 ) .
Examples
Understanding perpendicular lines is crucial in architecture and construction. For example, when designing a building, ensuring that walls are perpendicular to the ground is essential for stability. If a surveyor needs to determine the position of a wall that must be perpendicular to a reference line and pass through a specific point, they would use the principles of perpendicular lines to accurately mark the wall's location. This ensures the structural integrity and safety of the building.
The point on the line that is perpendicular to the given line and passes through the point on the y-axis is (0, -2). This is identified by analyzing the slopes of the lines and applying the properties of perpendicularity. Therefore, the correct answer from the options provided is (0, -2).
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