Final answer:
The equation of the perpendicular bisector of the line segment joining the points (7,1) and (3,5) can be found by first calculating the midpoint and slope of the line segment. The slope of the perpendicular line is the negative reciprocal of the original slope. We then use these values in the slope-intercept equation of a line to solve for the equation of the perpendicular bisector.
Step-by-step explanation:
The subject at hand lies in the realm of Geometry, a branch of Mathematics. To find the equation of the perpendicular bisector of the line segment joining the points (7,1) and (3,5), we first need to find the midpoint of the line segment and the slope of the original line.
The midpoint (M) can be found using the formula [(x1 + x2)/2 , (y1 + y2)/2]. Substituting the given coordinates in the formula, we get M = [(7 + 3) / 2 , (1 + 5) / 2] = (5,3).
The slope (m) of the line segment connecting (7,1) and (3,5) can be found using the formula (y2 - y1) / (x2 - x1); hence, m = (5 - 1) / (3 - 7) = -1. The slope of the perpendicular line will be the negative reciprocal of the original slope, which gives us 1.
Finally, the equation of the straight line can be written in slope-intercept form y = mx + b. We know the slope is 1 and the line passes through the point (5,3), hence if we substitute these values into the equation we can solve for b to find the equation of the perpendicular bisector.
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