at what value of x does the perpendicular bisector of the line segment with endpoints (-2,6) and (5,-4) cross the x-axis?
find the equation of the straight line between the two given points in the usual way, then
in that equation, let y = 0 and solve for x
Given:(-2,6),(5,-4). Mid-point:
Xo=(-2+5)/2=3/2, Yo=(6-4)/2=1,M(3/2,1),
Slope=(-4-6)/(5+2)=-7/10, Slope of
perpendicular bisector=7/10, Y=mX+b,
1=7/10*(3/2)+b, b=-1/20,Y=7/10X-1/20,
Y=7/10X-1/20=0,X=1/14=X-Intercept.
To find the value of x where the perpendicular bisector of the line segment crosses the x-axis, we need to follow these steps:
1. Find the midpoint of the line segment with endpoints (-2,6) and (5,-4).
Midpoint formula:
x-coordinate of midpoint = (x₁ + x₂) / 2
y-coordinate of midpoint = (y₁ + y₂) / 2
Applying the formula:
x-coordinate of midpoint = (-2 + 5) / 2 = 3/2 = 1.5
y-coordinate of midpoint = (6 + (-4)) / 2 = 2/2 = 1
So, the midpoint is (1.5, 1).
2. Find the slope of the line segment.
Slope formula:
slope = (y₂ - y₁) / (x₂ - x₁)
Applying the formula:
slope = (-4 - 6) / (5 - (-2)) = -10 / 7
So, the slope of the line segment is -10/7.
3. Determine the negative reciprocal of the slope to get the slope of the perpendicular bisector.
The negative reciprocal is the negative of the reciprocal, which means we flip the fraction and change the sign.
The reciprocal of -10/7 is -7/10.
The negative of -7/10 is 7/10.
So, the slope of the perpendicular bisector is 7/10.
4. Use the point-slope form of a line to find the equation of the perpendicular bisector.
Point-slope form:
y - y₁ = m(x - x₁)
Since we have the midpoint (1.5, 1) and the slope 7/10, we can substitute these values into the equation.
y - 1 = (7/10)(x - 1.5)
Simplifying the equation:
y - 1 = (7/10)x - (7/10)(1.5)
y - 1 = (7/10)x - 7/20
y = (7/10)x - 7/20 + 1
y = (7/10)x - 7/20 + 20/20
y = (7/10)x - 7/20 + 20/20
y = (7/10)x + 13/20
So, the equation of the perpendicular bisector is y = (7/10)x + 13/20.
5. Find the x-intercept of the perpendicular bisector.
To find the x-intercept, we set y = 0 and solve for x.
Setting y = 0 in the equation:
0 = (7/10)x + 13/20
Simplifying the equation:
(7/10)x = -13/20
Multiplying both sides by 10/7 to isolate the x-term:
x = (-13/20)(10/7)
x = -13/14
So, the x-intercept of the perpendicular bisector is -13/14.