The line represented by the equation y = 4 – 2x is the perpendicular

That's perfectly valid approach.

Two lines are perpendicular if and only the product of their slopes is -1. The slope of given line is -2, hence the slope of PR must be 1/2 (negative reciprocal of -2): 1/2*(-2)=-1.

Now, the slope of a line (a line segment) passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2-y_1}{x_2-x_1}\).

So, for our case the slope of PR must be \(m=\frac{1}{2}=\frac{1-y_1}{4-x_1}\) and you can substitute x and y coordinates of each point from answer choices to see for which one this equation will hold true. Only coordinates of a point from option D fits.

let's have a look
Required slope for line PR = m= 1/2 ( using if two lines are perpendicular then their slopes m1 * m2 = -1 )
option D= ( 0,-1)
Option E = ( 2, 0)

so taking (4,1) and ( 0,-1) and finding slope
= -2/-4 = 1/2 which is of course what we are expecting ,

now taking (4,1) and (2,0) and finding slope

= -1/-2 = 1/2

So also E, satisfies the slope method
So what am I missing , it is said that only one option satisfies the equation 1/2 = (1-y1)/( 4-x1)
Taking E( 2,0)
(1-0)/(4-2) = 1/2

and taking D(0,-1)
(1+1)/(4-0)= 2/4= 1/2

so both D and E satisfy the slope condition , am I missing anything ?

You can rule out (2,0) (option E), since this point is on the line y=4-2x.

Bunuel Bro, my question still remains : I understand E is on the line but why does E also result in slope of 1/2 . . E should cause the slope to be -2 since it is on the line. . why does it then cause the slope to be 1/2?

just trying to understand the math of slope here.

Point (2, 0) is on line segment PR (see diagram in my post above). PR is perpendicular to line y = 4 – 2x, thus ANY two point from line segment PR will give you the slope which is negative reciprocal of the slope of line y = 4 – 2x, i.e. 1/2.

Concrete way is to find the actual co-ordinates.

Slope of RP = 1/2 and by using 4,1 and slope the equation is y=x/2 -1

solve for point of intersection - x,y = 2,0.

now let P has co-ordinates (x,y)

since intersection is mid point.

(x+4)/2 = 2 and (y+1)/2 = 0 thus D

How do you get to know the point of intersection is 2,0?

I understood your solution till the previous step in which you found the equation of the line.

x-intercept is a value of x for y=0 and similarly y-intercept is a value of y for x=0.

To find x-intercept substitute y=0 and find x.
To find y-intercept substitute x=0 and find y.

For more check Coordinate Geometry chapter of Math Book: math-coordinate-geometry-87652.html

Hope it helps.

Bunuel, do you think these coordinates are not unique.
Any combination (x,y) for point P that satisfies the equation 2-2*y = 4-x will make RP perpendicular to the line y = 4-2x.
Am I correct to assert that?
Brother Karamazov

No, that's not correct. Line y = 4 – 2x not only has to be perpendicular of PR but also has to be bisector of PR (line y = 4 -2x cuts PR into two equal parts at 90°). Therefore the coordinates of P are unique.

Hope it's clear.

If the line y = 4 - 2x is the perpendicular bisector of line segment RP, then it’s perpendicular to RP at the midpoint of RP. Recall that two lines are perpendicular if their slopes are negative reciprocals of each other. Since the line has a slope of -2, the line segment should have a slope of ½. Thus, let’s first determine which answer choice can be point P so that the slope of RP is ½. We’ll use the slope formula: m = (y2 - y1)/(x2 - x1):

A) (-4, 1)

m = (1 - 1)/(4 - (-4)) = 0

Point P can’t be (-4, 1).

B) (-2, 2)

m = (1 - 2)/(4 - (-2)) = -1/6

Point P can’t be (-2, 2).

C) (0, 1)

m = (1 - 1)/(4 - 0) = 0

Point P can’t be (0, 1).

D) (0, -1)

m = (1 - (-1))/(4 - 0) = 2/4 = 1/2

Point P could be (0, -1).

E) (2, 0)

m = (1 - 0)/(4 - 2) = 1/2

Point P could be (2, 0).

We see that point P could be either (0, -1) or (2, 0), since either one will make RP’s slope ½. Next let’s determine the midpoint of RP if P is either (0, -1) or (2, 0). We use the midpoint formula: ((x1 + x2)/2 , (y1 + y2)/2)). Recall that R = (4,1).

If P = (0, -1), then the midpoint of RP = ((4 + 0)/2, (1 + (-1))/2) = (2, 0).

If P = (2, 0), then the midpoint of RP = ((4 + 2)/2, (1 + 0)/2) = (3, 1/2).

Recall that the line y = 4 - 2x has to include the midpoint of RP. In other words, the midpoint of RP is a point on the line, and hence its coordinates will satisfy the equation of the line.

If P = (0, -1) and the midpoint of RP is (2, 0), is 0 = 4 - 2(2)? The answer is yes, since 0 = 0. We can see that choice D is the correct choice. However, let’s also show that choice E is not the correct choice:

If P = (2, 0) and the midpoint of RP is (3, 1/2), is 1/2 = 4 - 2(3)? The answer is no, since 1/2 ≠ -2.

Alternate Solution:

Since the line segment RP is perpendicular to y = 4 - 2x, the line containing RP must have a slope of 1/2, since the slopes of perpendicular lines are negative reciprocals of each other. Then, the line containing the line segment RP must be of the form y = (1/2)x + b, for some real number b. Since this line contains the point R, its equation must be satisfied when we substitute x = 4 and y = 1; therefore 1 = (1/2)(4) + b. Then, 1 = 2 + b, and thus b = -1.

We know the line that contains the line segment RP has an equation y = (1/2)x - 1. Let’s find the common point of this line with y = 4 - 2x. We set (1/2)x - 1 = 4 - 2x and solve for x:

(1/2)x -1 = 4 - 2x

(5/2)x = 5

x = 2

So, the x-coordinate of the common point is 2. We can substitute x = 2 in either equation to find the y-coordinate: y = 4 - 2(2) = 0. So, the common point is (2, 0).

Note that this common point is the midpoint of R and P. Therefore, if we let P = (a, b), we must have:

(a + 4)/2 = 2 and (b + 1)/2 = 0. We find that a = 0 and b = -1. Thus, P = (0, -1).

Answer: D

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Equation of Perpendicular Bisector is
Y=4-2X
Equation of a line perpendicular to this line
Y=1/2X+C
This line passes through (4,1)
1=1/2 *4 +C
C=-1
Equation of line through which perpendicular bisector passes or equation of line RP is
Y=1/2 X-1
Coordinates of point P must pass through this Equation
At X=0, Y=-1
So (0,-1) But (2,0) also satisfies this condition.
If (2,0) is the coordinates of point P
Then Mid point of RP(3,1/2), and this must be pass through the line of perpendicular bisector
i.e
Y=4-2X
at X =3, Y=-2( Not 1/2)
So (2,0) is not the coordinates of point P
D:)

for eqn y = 4 – 2x
slope= -2
so for line with coordiante (4,1) the slope should be 1/2
since both are perpendicular so m1*m2=-1
we get 1/2 for (4,1) & ( 0,-1) only
IMO D

slope of line y=4-2x is -2
slope of RP = 1/2 (perpendicular bisector)
To find equation of line RP y = mx+c
put m = 1/2 y=1 and x=4 to find C (Rs coordinates given 4,1)
upon solving, c=-1

equation of line RP is y=1/2x- 1
Point P lies on this line and hence will have to satisfy the equation
ONLY points (0,-1) satisfies this equation
-1=-1 hence D

Hi Bunuel, I followed a similar approach but did not solve by plotting the points to scale for calculating R. Instead i figured that the R must be 180 degree opposite to point P and should also be 90 degree opposite the line (meaning the simplified line equation y = -x acts as a mirror ). Therefore, the point R should lie in 3rd or 4th quadrant. Only 1 option had negative sign for y coordinate and i picked that as my answer.

I want to know if this reasoning is correct? (i know if there are multiple options with negative y coordinates, this approach will not work)

IMO D

Given:
y = 4 - 2x
can be written as
y = -2x + 4
so slope = -2 ---> slope of perpendicular line RP = 1/2

Slope point form: m = 1/2 ; Point (4,1)
Substitute choices in m = (y2-y1) / (x2-x1) to get 1/2

So, answer is D