Let the line p be the perpendicular bisector of A = (24, 7) and B = (3, 4). Given that AB meets p at C = (x, y), what is 2x - 4y?
perpendicular bisector of AB crosses at the midpoint.
The midpoint of AB is (27/2 , 11/2)
2x-4y = 27 - 22 = 5
Clearly the perpendicular bisector has to meet AB at the midpoint of AB, so
you need the midpoint of AB which is
C( (24+3)/2 , (7+4)/2 ) = C(27/2 , 11/2)
that is x = 27/2 and y = 11/2
2x - 4y
= 27 - 22 = 5
To find the value of 2x - 4y, we first need to find the coordinates of point C where line AB intersects the perpendicular bisector of AB.
Step 1: Find the midpoint of AB.
The midpoint of AB is calculated by taking the average of the x-coordinates and the average of the y-coordinates of A and B, respectively.
Midpoint formula: ( (x₁ + x₂) / 2, (y₁ + y₂) / 2 )
Midpoint of AB = ( (24 + 3) / 2, (7 + 4) / 2 )
= ( 27 / 2 , 11 / 2 )
= ( 13.5 , 5.5 )
Step 2: Find the slope of AB.
The slope of AB is calculated using the slope formula: (y₂ - y₁) / (x₂ - x₁).
Slope of AB = (4 - 7) / (3 - 24)
= -3 / -21
= 1 / 7
Step 3: Find the negative reciprocal of the slope.
Since the perpendicular bisector of AB is perpendicular to AB, its slope will be the negative reciprocal of the slope of AB. To find the negative reciprocal, flip the fraction and change its sign.
Negative reciprocal of 1/7 = -7/1 = -7
Step 4: Use the midpoint and the negative reciprocal slope to find the equation of the perpendicular bisector.
The equation of a line can be expressed in the form y = mx + b, where m is the slope and b is the y-intercept.
Using the midpoint (x₁, y₁) = (13.5, 5.5) and the slope m = -7, we can calculate the y-intercept b using the formula: b = y - mx.
b = 5.5 - (-7)(13.5)
= 5.5 + 7(13.5)
= 5.5 + 94.5
= 100
The equation of the perpendicular bisector line p is: y = -7x + 100.
Step 5: Find the coordinates of point C by solving the equations of lines AB and p.
To find the coordinates of point C, we need to solve the system of equations formed by the equations of lines AB and p:
AB: y = (1/7)x + b₁
p: y = -7x + b₂
Setting the two equations equal to each other, we have:
(1/7)x + b₁ = -7x + b₂
Since both equations have the same y-intercept (b₁ = b₂ = 100), we can set them equal to each other:
(1/7)x + 100 = -7x + 100
Simplifying the equation:
(1/7)x + 7x = 0
Multiplying both sides of the equation by 7 to eliminate the fractions:
x + 49x = 0
Combining the x-terms:
50x = 0
Dividing both sides by 50:
x = 0
Plugging the value of x back into either equation:
y = -7(0) + 100
y = 100
Therefore, the coordinates of point C are (0, 100).
Step 6: Calculate 2x - 4y:
2x - 4y = 2(0) - 4(100)
= 0 - 400
= -400
Therefore, 2x - 4y = -400.