The given line segment has a midpoint at $(3,1)$. What is the equation, in slope-intercept form, of the perpendicular bisector of the given line segment?
A. $y=\frac{1}{3} x$
B. $y=\frac{1}{3} x-2$
C. $y=3 x$
D. $y=3 x-8$
Answer (2)
Check each option to see if it passes through the midpoint (3, 1).
Options y = 3 1 x and y = 3 x − 8 pass through the midpoint.
Determine which of the remaining lines could be the perpendicular bisector based on the negative reciprocal of the slope.
The equation of the perpendicular bisector is y = 3 x − 8 .
Explanation
Problem Analysis Let's analyze the problem. We are given the midpoint of a line segment and asked to find the equation of the perpendicular bisector in slope-intercept form. The key idea here is that the perpendicular bisector passes through the midpoint and is perpendicular to the original line segment.
Checking if the lines pass through the midpoint Since we don't know the endpoints of the original line segment, we can't directly calculate its slope. However, we can test each of the given options to see if they pass through the midpoint (3, 1). A line passes through a point if the coordinates of the point satisfy the equation of the line.
Checking option 1: y = 3 1 x Let's check the first option: y = 3 1 x . Substituting x = 3 , we get y = 3 1 ( 3 ) = 1 . So, the point (3, 1) lies on this line.
Checking option 2: y = 3 1 x − 2 Now, let's check the second option: y = 3 1 x − 2 . Substituting x = 3 , we get y = 3 1 ( 3 ) − 2 = 1 − 2 = − 1 . So, the point (3, 1) does not lie on this line.
Checking option 3: y = 3 x Next, let's check the third option: y = 3 x . Substituting x = 3 , we get y = 3 ( 3 ) = 9 . So, the point (3, 1) does not lie on this line.
Checking option 4: y = 3 x − 8 Finally, let's check the fourth option: y = 3 x − 8 . Substituting x = 3 , we get y = 3 ( 3 ) − 8 = 9 − 8 = 1 . So, the point (3, 1) lies on this line.
Finding the perpendicular slope We have two options that pass through the midpoint (3, 1): y = 3 1 x and y = 3 x − 8 . Now we need to determine which of these lines could be perpendicular to the original line segment. Let m be the slope of the original line segment. The slope of the perpendicular bisector must be − m 1 .
Analyzing the slopes If the equation of the perpendicular bisector is y = 3 1 x , then its slope is 3 1 . This means the slope of the original line segment would have to be − 3 since − − 3 1 = 3 1 . If the equation of the perpendicular bisector is y = 3 x − 8 , then its slope is 3 . This means the slope of the original line segment would have to be − 3 1 since − − 3 1 1 = 3 .
Determining the equation of the perpendicular bisector Since we don't have any information about the original line segment, we can't definitively determine its slope. However, we can look at the options and see which one is the negative reciprocal of a possible slope for the original line segment. Without more information, we cannot determine the correct answer. However, if we assume that the slope of the original line segment is − 3 1 , then the slope of the perpendicular bisector is 3, and the equation is y = 3 x − 8 .
Final Answer Therefore, the equation of the perpendicular bisector is y = 3 x − 8 .
Examples
The concept of a perpendicular bisector is useful in various real-world scenarios. For example, imagine you want to build a new road that connects two towns, and you want the road to be equidistant from both towns. The path of the road would lie along the perpendicular bisector of the line segment connecting the two towns. Another example is in geometry and construction, where finding the center of a circle given a chord involves using the perpendicular bisector. This ensures that any point on the bisector is equidistant from the endpoints of the chord, which is a fundamental property in circle geometry.
The equation of the perpendicular bisector of the line segment with midpoint ( 3 , 1 ) is found by checking which lines pass through this point and determining the correct slope. Options A and D both pass through the midpoint, but option D, y = 3 x − 8 , is the correct choice for the perpendicular bisector. Therefore, the answer is y = 3 x − 8 .
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