Asked by Cons
A diagram shows lines x-2y-4=0, x+ y=5 and the point P (1,1). The line is drawn from P to intersect with x-2y-4= 0 at Q, and with x+y=5 at point R, so the P is midpoint of the QR. Find the coordinates for the point R and Q.
Answers
Answered by
Bosnian
a = x coordinate of point Q on line x - 2 y - 4 = 0
b = y coordinate of point Q on line x - 2 y - 4 = 0
Put this coordinates in equation:
x - 2 y - 4 = 0
a - 2 b - 4 = 0
Add 4 to both sides
a - 2 b = 4
If P is the midpoint of QR, then:
( R + Q ) / 2 = P
Multiply both sides by 2
R + Q = 2 P
Subtract Q to both sides
R = 2 P - Q
x coordinate of point P = 1
y coordinate of point P = 1
x coordinate of point Q = a
y coordinate of point Q = b
R = 2 P - Q
R = 2 ( 1 , 1 ) - ( a , b ) = ( 2 , 2 ) - ( a , b ) = ( 2 - a ) , ( 2 - b )
You know this point also satisfies the equation:
x + y = 5
Put x coordinate of point x = 2 - a and y coordinate of point y = 2 - b in this equation:
( 2 - a ) + ( 2 - b ) = 5
2 - a + 2 - b = 5
4 - a - b = 5
Subtract 4 to both sides
- a - b = 1
Multiply both sides by - 1
a + b = - 1
To find the solution you can subtract the first from the second:
( a + b = - 1 ) - ( a - 2 b = 4 )
a + b - ( a - 2 b ) = - 1 - 4
a + b - a + 2 b = - 5
3 b = - 5
b = - 5 / 3
Now:
a + b = - 1
Subtract b a to both sides
a = - 1 - b
a = - 1 - ( - 5 / 3 ) = - 3 / 3 + 5 / 3 = 2 / 3
The point Q is ( a , b ) = ( 2 / 3, - 5 / 3 )
The point R is:
R = ( 2 - a , 2 - b ) = ( 2 - 2 / 3, ) , [ 2 - ( - 5 / 3 ) ] =
( 6 / 3 - 2 / 3 , 2 + 5 / 3 ) = ( 4 / 3 , 6 / 3 + 5 / 3 ) = ( 4 / 3 , 11 / 3 )
The coordinates of Q and R are:
Q ( 2 / 3, - 5 / 3 )
R ( 4 / 3, 11 / 3 )
b = y coordinate of point Q on line x - 2 y - 4 = 0
Put this coordinates in equation:
x - 2 y - 4 = 0
a - 2 b - 4 = 0
Add 4 to both sides
a - 2 b = 4
If P is the midpoint of QR, then:
( R + Q ) / 2 = P
Multiply both sides by 2
R + Q = 2 P
Subtract Q to both sides
R = 2 P - Q
x coordinate of point P = 1
y coordinate of point P = 1
x coordinate of point Q = a
y coordinate of point Q = b
R = 2 P - Q
R = 2 ( 1 , 1 ) - ( a , b ) = ( 2 , 2 ) - ( a , b ) = ( 2 - a ) , ( 2 - b )
You know this point also satisfies the equation:
x + y = 5
Put x coordinate of point x = 2 - a and y coordinate of point y = 2 - b in this equation:
( 2 - a ) + ( 2 - b ) = 5
2 - a + 2 - b = 5
4 - a - b = 5
Subtract 4 to both sides
- a - b = 1
Multiply both sides by - 1
a + b = - 1
To find the solution you can subtract the first from the second:
( a + b = - 1 ) - ( a - 2 b = 4 )
a + b - ( a - 2 b ) = - 1 - 4
a + b - a + 2 b = - 5
3 b = - 5
b = - 5 / 3
Now:
a + b = - 1
Subtract b a to both sides
a = - 1 - b
a = - 1 - ( - 5 / 3 ) = - 3 / 3 + 5 / 3 = 2 / 3
The point Q is ( a , b ) = ( 2 / 3, - 5 / 3 )
The point R is:
R = ( 2 - a , 2 - b ) = ( 2 - 2 / 3, ) , [ 2 - ( - 5 / 3 ) ] =
( 6 / 3 - 2 / 3 , 2 + 5 / 3 ) = ( 4 / 3 , 6 / 3 + 5 / 3 ) = ( 4 / 3 , 11 / 3 )
The coordinates of Q and R are:
Q ( 2 / 3, - 5 / 3 )
R ( 4 / 3, 11 / 3 )
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