write an equation in general form...
the perpendicular bisector of the segment with endpoints at (2,-1) and (4,-5). I found the slope and then found the mid point and wrote to the equation and got X-2Y=3. But the answer is X-2Y=9...any help???
slope is of the segment is -2, you got that correct, and slope of bisector is 1/2, right. The midpoint is at 3,-3
y=mx+b
-3=3/2 + b
b= -4.5
y=1/2 x -4.5
2y=x-9
x-2y=9
The mid point of the segment is
((x1+x2)/2, (y1+y2)/2)=(3,-3)
You have correctly calculated the slope m as 1/2.
Using the equation
(y-y1)=m(x-x1),
we get
(y-(-3))=(1/2)(x-3)
2(y+3)=x-3
x-2y=9
You have probably mis-calculated the mid-point as (-3,-3)
To find the equation of the perpendicular bisector of a segment, you need to follow these steps:
Step 1: Find the midpoint of the segment.
Step 2: Find the negative reciprocal of the slope of the segment.
Step 3: Use the midpoint and the negative reciprocal slope to write the equation in point-slope form.
Step 4: Convert the equation to general form.
Let's go through these steps:
Step 1: Find the midpoint
The midpoint of the segment with endpoints (2, -1) and (4, -5) can be found using the mid-point formula:
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Plugging in the values:
Midpoint = ((2 + 4)/2, (-1 + -5)/2)
= (6/2, -6/2)
= (3, -3)
So, the midpoint of the segment is (3, -3).
Step 2: Find the negative reciprocal of the slope
To find the negative reciprocal of the slope, you need to take the negative value of the reciprocal of the slope of the segment.
The slope of the segment can be found using the slope formula:
Slope = (y₂ - y₁)/(x₂ - x₁)
Plugging in the values:
Slope = (-5 - (-1))/(4 - 2)
= (-5 + 1)/(4 - 2)
= -4/2
= -2
The negative reciprocal of -2 is 1/2.
Step 3: Write the equation in point-slope form
Using the midpoint (3, -3) and the negative reciprocal slope of 1/2, we can write the equation of the perpendicular bisector in point-slope form:
y - y₁ = m(x - x₁)
Plugging in the values:
y - (-3) = 1/2(x - 3)
y + 3 = 1/2x - 3/2
Step 4: Convert the equation to general form
To convert the equation to general form (Ax + By = C), we need to eliminate fractions and rearrange the equation:
Multiply the entire equation by 2 to eliminate the fraction:
2(y + 3) = 2(1/2x - 3/2)
2y + 6 = x - 3
Rearrange the equation:
x - 2y = 3 - 6
x - 2y = -3
So, the equation of the perpendicular bisector of the segment with endpoints (2, -1) and (4, -5) should be x - 2y = -3, not x - 2y = 9.