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

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The line represented by the equation y = 4 – 2x is the [#permalink] New post 18 Feb 2012, 17:55
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The line represented by the equation y = 4 – 2x is the perpendicular bisector of line segment RP. If R has the coordinates (4, 1), what are the coordinates of point P?

(A) (–4, 1)
(B) (–2, 2)
(C) (0, 1)
(D) (0, –1)
(E) (2, 0)

[Reveal] Spoiler:
How come the answer will be D? This is how I am trying to solve this.

First, rewrite the line y=4-2x as y = -2x+4 The equation is now in the form y = mx+b where m represents the slope and b represents the y-intercept.Thus, the slope of this line is -2. By definition, if a line is the perpendicular bisector of any line, the slope of line which is perpendicular bisector is the negative inverse of the slope of line G. Since we are told that the line y = -2x+4 is the perpendicular bisector of line segment RP, line segment RP
must have a slope of \frac{1}{2} (which is the negative inverse of slope of line y).
Now we know that the slope of the line containing segment RP is\frac{1}{2} but we do
not know its y-intercept. We can write the equation of this line as , y = 1/2x+b, where b represents the unknown y-intercept.
To solve for b, we can use the given information that the coordinates of point R
are (4, 1). Since point R is on the line y = 1/2x+b, we can plug 4 in for x and 1 in for y to get b = -1
Therefore, equation of line RP will become y = 1/2x-1
Also , y = -2x +4 (Equation of perpendicular bisector) -----------------(2)

Equating the two we will get x =2 . Putting this value of x in we get y = 0.

So the points should be (2,0) i.e. answer E.

Where I am getting this wrong guys?
[Reveal] Spoiler: OA

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Re: Coordinates of Point P [#permalink] New post 18 Feb 2012, 18:12
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The line represented by the equation y = 4 – 2x is the perpendicular bisector of line segment RP. If R has the coordinates (4, 1), what are the coordinates of point P?

A. (–4, 1)
B. (–2, 2)
C. (0, 1)
D. (0, –1)
E. (2, 0)

Again, there is no need of equations to solve this question. Plot the line y = 4 – 2x (just find the x and y intercepts and draw the line through them):
Attachment:
Bisector.png
Bisector.png [ 16 KiB | Viewed 10138 times ]
Now, it's easy to SEE that no blue point can be the mirror reflection of R around the line but (0, -1).

Answer: D.

P.S. Answer cannot possibly be E (2, 0) as this point lies on the line y=4-2x (substitute the values of x and y to see that it's true).
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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 25 Feb 2012, 06:34
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Hello,

This is how I solved the problem. Since the Slope of line is -2, the line perpendicular to it would have SLope as 1/2. So I used R(4,1) and each options to see which one gives Slope as 1/2. Only option D gives me the co ordinates through which the SLope is 1/2. It took me around a minutes time to solve.

Please let me know if I'm correct ?
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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 25 Feb 2012, 08:30
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priyalr wrote:
Hello,

This is how I solved the problem. Since the Slope of line is -2, the line perpendicular to it would have SLope as 1/2. So I used R(4,1) and each options to see which one gives Slope as 1/2. Only option D gives me the co ordinates through which the SLope is 1/2. It took me around a minutes time to solve.

Please let me know if I'm correct ?


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.
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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 01 Jul 2012, 02:10
Bunuel wrote:
priyalr wrote:
Hello,

This is how I solved the problem. Since the Slope of line is -2, the line perpendicular to it would have SLope as 1/2. So I used R(4,1) and each options to see which one gives Slope as 1/2. Only option D gives me the co ordinates through which the SLope is 1/2. It took me around a minutes time to solve.

Please let me know if I'm correct ?


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 ?
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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 01 Jul 2012, 02:37
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stne wrote:
Bunuel wrote:
priyalr wrote:
Hello,

This is how I solved the problem. Since the Slope of line is -2, the line perpendicular to it would have SLope as 1/2. So I used R(4,1) and each options to see which one gives Slope as 1/2. Only option D gives me the co ordinates through which the SLope is 1/2. It took me around a minutes time to solve.

Please let me know if I'm correct ?


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.
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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 02 Oct 2012, 07:19
ok but why on earth does E also result in slope oh 1/2 ? :shock:
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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 02 Oct 2012, 07:40
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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 26 Oct 2012, 05:29
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.
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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 26 Oct 2012, 05:41
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sachindia wrote:
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.
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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 26 Oct 2012, 08:52
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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
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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 01 Nov 2012, 23:45
Quote:
solve for point of intersection - x,y = 2,0.


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.
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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 02 Nov 2012, 01:07
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Quote:
solve for point of intersection - x,y = 2,0.


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.
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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 11 Nov 2012, 07:08
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?
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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 12 Nov 2012, 09:02
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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.
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The line represented by the equation y = 4 – 2x is the [#permalink] New post 13 Nov 2012, 02:12
I will go for D
slope of MP (with P(a, b)) should be 1/2
thus 1-b/4-a = 1/2 2-2*b = 4-a
only D satisfies this equation
But Bunuel, if there were no answers choices, how one could find these values for a and b?
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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 13 Nov 2012, 02:22
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Ousmane wrote:
I will go for D
slope of MP (with P(a, b)) should be 1/2
thus 1-b/4-a = 1/2 2-2*b = 4-a
only D satisfies this equation
But Bunuel, if there were no answers choices, how one could find these values for a and b?


Merging similar topics. Please refer to the solutions above.
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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 09 Sep 2013, 20:18
Hi Bunuel, You drew a descending slope in forward direction that is confusing me, slope should be descending is understandable because m = -2 but why it is in forward direction now backward , i mean why is it not passing through quadrant 3.??

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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 10 Sep 2013, 00:33
Expert's post
ishdeep18 wrote:
Hi Bunuel, You drew a descending slope in forward direction that is confusing me, slope should be descending is understandable because m = -2 but why it is in forward direction now backward , i mean why is it not passing through quadrant 3.??

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Ishdeep Singh


I don't understand what does "descending slope in forward direction" is.
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Re: The line represented by the equation y = 4 – 2x is the [#permalink] New post 12 Sep 2013, 16:52
Hello Bunuel,

Request you to please validate my approach:

We know that perpendicular bisector bisects the line into two equal halves.
Since, line given is the perpendicular bisector of line segment RP. hence, point M will be the midpoint of line segment RP,.
Using mid point formula
Let point of P(x,y)
So,
(x+4)/2=2

(y+1)/2 = 0

Hence, coordinates are (0,-1).

Is this a valid approach or answer just happens to be correct/.
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Re: The line represented by the equation y = 4 – 2x is the   [#permalink] 12 Sep 2013, 16:52
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