Answer: a) To find the equation of line PS, we first need to find the slope of the line L1 which is given by:
slope = (y2 - y1) / (x2 - x1)
= (3 - 5) / (-2 - 0)
= 2/2
= 1
Since PS is parallel to L1, it also has the same slope. We can now use the point-slope form of the equation of a line to find the equation of PS, using the point P(1,2):
y - y1 = m(x - x1)
y - 2 = 1(x - 1)
y - x + 2 = 0
Therefore, the equation of line PS is y - x + 2 = 0.
b) To find the coordinates of S, we know that S lies on line PS and also on the line passing through points (6,4) and P(1,2). Let's call the coordinates of S as (x,y). Since S lies on line PS, we know that y - x + 2 = 0. We can also use the slope formula to find the slope of line PS:
slope_PS = (y - 2) / (x - 1)
Since PS is parallel to line L1, we know that slope_PS = 1. Therefore, we can write:
(y - 2) / (x - 1) = 1
y - 2 = x - 1
y = x + 1
Now, we can use the fact that S also lies on the line passing through points (6,4) and P(1,2). This line has the equation:
(y - 4) / (x - 6) = (2 - 4) / (1 - 6)
(y - 4) / (x - 6) = -2/-5
(y - 4) / (x - 6) = 2/5
We can solve for y in terms of x:
y - 4 = (2/5)(x - 6)
y = (2/5)x - 8/5 + 4
y = (2/5)x + 6/5
Now, we can set the two equations for y equal to each other, since they both represent the y-coordinate of point S:
x + 1 = (2/5)x + 6/5
Solving for x, we get:
x = 2
Substituting x = 2 into either of the equations for y, we get:
y = 3
Therefore, the coordinates of S are (2,3).
c) To find the coordinates of Q, we can use the fact that PQ is parallel to SR and PS is parallel to QR. This means that PQRS is a parallelogram and its opposite sides are parallel. Since we know the coordinates of P and S, we can find the coordinates of Q and R by adding or subtracting the appropriate vector from P and S.
The vector that we need to add to P to get Q is the same vector that we need to add to S to get R, which is given by the displacement vector PS = <5,1>. Therefore:
Q = P + PS
= <1,2> + <5,1>
= <6,3>
Therefore, the coordinates of Q are (6,3).
d) To find the equation of line L2 which is the perpendicular bisector of line QR, we first need to find the midpoint of QR. The midpoint formula is given by:
midpoint = [(x1 +)]
We can then find the midpoint of QR using the midpoint formula:
midpoint of QR = ((-4 + 1)/2, (7 + 4)/2) = (-1.5, 5.5)
So the slope of L2 is the negative reciprocal of the slope of QR:
slope of L2 = -1/slope of QR = -(7-5)/(1+4) = -2/5
We can use the point-slope form of the equation of a line to find the equation of L2. We know that L2 passes through the midpoint of QR, so we can use the point (-1.5, 5.5):
y - 5.5 = (-2/5)(x + 1.5)
Simplifying and rearranging, we get:
y = (-2/5)x + 7.5
Therefore, the equation of line L2 is y = (-2/5)x + 7.5.