B (5,2).
the slope of this line is (y2 - y1) / (x2 - x1), where (x1,y1) = (-3,4)and (x2,y2) = (5,2).
the slope is therefore (2-4) / (5-(-3)) = -2/8 = -1/4
the equation of the line, in slope intercept form, would be y = mx + b.
m is the slope.
b is the y-intercept.
replace m with -1/4 and the equation becomes y = -1/4 * x + b
replace x and y with the value of one of the points that the line goes through and the equation becomes:
2 = -1/4 * 5 + b, if using (5,2)
solve for b to get:
b = 2 + 1/4 * 5, resulting in b = 3.25
the equation becomes y = -1/4 * x + 3.25
here's the graph, showing it going through all the points that were given and calculated algebraically.
the graph shows the x-intercept as well as the y-intercept.
the line segment starts at (-4,3) and ends at (5,2).
the midpoint of this line seqment will be h = ((x1+x2)/2,(y1+y2)/2), where h represents the midpoint.
replace (x1,x2) with (-3,4) and (x2,y2) with (5,2) and the equation becomes ((-3+5)/2,(4+2)/2).
this results in h = (1,3).
the slope of the line perpendicular to the original line will have a slope that is a negative reciprocal of the original line.
the slope of the original line is -1/4.
the slope of the line perpendicular to it will be 4.
the slope intercept form of the perpendicular equation will be y = 4x + b.
replace (x,y) with (1,3) and the equation becomes:
3 = 4*1 + b
solve for b to get:
b = -1
the equation of the perpendicular line going through the point (1,3) is y = 4x -1
here's the graph of the perpendicular line and the original line.
the line is perpendicular because its slope is a negative reciprocal of the original line.
the line is a bisector because the length of the line segments to the left of the point of intersection and the right of the point of intersection are congruent.
the formula for the length of a line is k = sqrt((x2-x1)^2 + (y2-y1)^2), where k represents the length of the line (the variable name is chosen arbitrarily and can be any variable name that isn't currently used in the equation).
the length of the line segment between (-3,4) and (1,3) is equal to sqrt((1-(-3))^2 + (3-4)^2) which becomes sqrt((4)^2 + (-1)^2) which becomes sqrt(16+1) which is equal to sqrt(17).
the length of the line segment between (1,3) and (5,2) is equal to sqrt((5-1)^2 + (2-3)^2) which becomes sqrt((4)^2 + (-1)^2) which becomes sqrt(16+1) which is equal to sqrt(17).
the lengths of the line segments are equal, therefore the point (1,3) is a bisector of the line between (-3,4) and (5,2).