The following diagram shows the lines x – 2y – 4 = 0, x + y = 5 and the point P(1, 1). A line is drawn from P to intersect with x – 2y – 4 = 0 at Q, and with x + y = 5 at R, so that P is the midpoint of [QR]. Find the exact coordinates of Q and of R.

Step-by-step explanation:

Let (a, b) represent the coordinates of point Q on line x -2y -4 = 0. Then we know that ...

  a -2b -4 = 0

  a -2b = 4

__

If P is the midpoint of QR, then ...

  R = 2P -Q = 2(1, 1) -(a, b) = (2 -a, 2 -b)

We know this point satisfies the equation for the other line:

  x + y = 5

  (2 -a) +(2 -b) = 5

  a + b = -1 . . . . . . . . . . . rearrange to standard form

__

To find the solution to these two equations, we can subtract the first from the second:

  (a +b) -(a -2b) = (-1) -(4)

  3b = -5

  b = -5/3

  a = -1 -1b = -1 -(-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)) = (4/3, 11/3)

The exact coordinates of Q and R are ...

  Q(2/3, -5/3), R(4/3, 11/3)

_____

Comment on the GeoGebra solution

The points can be found by reflecting either line across point P. Where that reflected line intersects the other line is one of the points of interest. Of course, the other is its reflection in P. You may recognize the equation for line b' (hidden in the diagram) as matching the second equation we derived above.

91

Step-by-step explanation:

dfsdvcxz ssvdvsd