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The line represented by the equation y = 4 – 2x is the [#permalink]

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18 Feb 2012, 18:55

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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?

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.

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 [ 16 KiB | Viewed 25076 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).
_________________

Re: The line represented by the equation y = 4 – 2x is the [#permalink]

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25 Feb 2012, 07:34

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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.

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.
_________________

Re: The line represented by the equation y = 4 – 2x is the [#permalink]

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01 Jul 2012, 03:10

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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 ?
_________________

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.
_________________

Re: The line represented by the equation y = 4 – 2x is the [#permalink]

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26 Oct 2012, 06:29

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.
_________________

hope is a good thing, maybe the best of things. And no good thing ever dies.

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.
_________________

Re: The line represented by the equation y = 4 – 2x is the [#permalink]

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11 Nov 2012, 08:08

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? Brother Karamazov

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.

Re: The line represented by the equation y = 4 – 2x is the [#permalink]

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09 Sep 2013, 21:18

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.??

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.??

Regards 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]

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12 Sep 2013, 17:52

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/.
_________________

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/.

HOW do you know that the midpoint of PR is (2, 0)?
_________________

Re: The line represented by the equation y = 4 – 2x is the [#permalink]

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13 Sep 2013, 04:53

I've actually used approach with a slope calculation, but when I looked at the answers I realized that there was a shortcut: there was only one answer in negative y semi-plane. And even very rough plot is enough to realize that it must be true for the correct answer, I believe.

Re: The line represented by the equation y = 4 – 2x is the [#permalink]

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14 Sep 2013, 16:43

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 ?

Slope of PR = 1/2; (1/2)x(-2) = -1 because y = 4 - 2x is perpendicular to PR (PR) y = 0.5x + k. Since R(4,1) is in PR, so: 1 = 0.5x4 + k, therefore k = -1 (PR) y = 0.5x - 1 intersect y = 4 - 2x at point D With y = 0.5x - 1 and y = 4 - 2x, we have point D (2,0) Since D is the midpoint of PR, Xr + Xp = 2Xd and Yr + Yp = 2Yd With Xr = 4, Yr = 1; Xd = 2, Yd = 0. So Xp = 0, Yp = -1. P(0,-1) The answer is D.

Re: The line represented by the equation y = 4 – 2x is the [#permalink]

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13 Apr 2014, 12:33

imhimanshu wrote:

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/.

For perpendicular bisectors m1m2= -1. here in line given slope is -2 so now for bisector it should be 1/2. we can pick the values from option one by one and get the slope