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# The two perpendicular lines L and R intersect at the point

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The two perpendicular lines L and R intersect at the point [#permalink]

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13 Nov 2013, 01:20
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The two perpendicular lines L and R intersect at the point (3, 4) on the coordinate plane to form a triangle whose other two vertices rest on the x-axis. If one of the other two vertices is located at the origin, what is the x-coordinate of the other vertex?

A. 11/8
B. 33/8
C. 5
D. 20/3
E. 25/3
[Reveal] Spoiler: OA

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Re: The two perpendicular lines L and R intersect at the point [#permalink]

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13 Nov 2013, 01:56
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honchos wrote:
The two perpendicular lines L and R intersect at the point (3, 4) on the coordinate plane to form a triangle whose other two vertices rest on the x-axis. If one of the other two vertices is located at the origin, what is the x-coordinate of the other vertex?

A. 11/8
B. 33/8
C. 5
D. 20/3
E. 25/3

Look at the diagram below:
Attachment:

Untitled.png [ 6.93 KiB | Viewed 3011 times ]

For a line that crosses two points $$(x_1,y_1)$$ and $$(x_2,y_2)$$, slope $$m=\frac{y_2-y_1}{x_2-x_1}$$

Thus the slope of line L passing (0, 0) and (3, 4) is $$m=\frac{y_2-y_1}{x_2-x_1}=\frac{4-0}{3-0}=\frac{4}{3}$$.

The slope of line R, which is perpendicular to L, is negative reciprocal of the slope of L, hence its $$-\frac{3}{4}$$.

Now, we need to find equation of R.

The equation of a straight line that passes through a point $$P_1(x_1, y_1)$$ with a slope m is: $$y-y_1=m(x-x_1)$$. Therefore, the equation of R is $$y-4=-\frac{3}{4}(x-3)$$ --> $$y=-\frac{3}{4}x+\frac{25}{4}$$. The x-intercept of this line is 25/3.

For more check here: math-coordinate-geometry-87652.html
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Re: The two perpendicular lines L and R intersect at the point [#permalink]

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16 Nov 2013, 12:55
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Hi honchos...you could even do this :-
as bunuel has calculated, the slope of the line will be -3/4. Now suppose the co-ordinates of the point needed are (a,0), just equate the slopes.
You will get the equation
-3/4 = 4/(3-a)
solving this you get the OA 25/3 i.e. E.
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Re: The two perpendicular lines L and R intersect at the point [#permalink]

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22 Aug 2015, 07:07
honchos wrote:
The two perpendicular lines L and R intersect at the point (3, 4) on the coordinate plane to form a triangle whose other two vertices rest on the x-axis. If one of the other two vertices is located at the origin, what is the x-coordinate of the other vertex?

A. 11/8
B. 33/8
C. 5
D. 20/3
E. 25/3

I think this can be solved using similar triangle properties (right angled)
Draw a perpendicular line from the top vertex to x-axis which intersects the axis at (3, 0)
You have one right angled triangle with sides 3, 4, 5
Use the properties to get the other sides.

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Re: The two perpendicular lines L and R intersect at the point [#permalink]

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21 Oct 2016, 09:00
Or you can use the distances:
To find the l side you can simply find the distance from (3,4) to (0,0) which is sqr[ (3-0)^2 +(4-0)^2 ] = 5.
The R side is the distance from (3,4) to (x,0) which is : sqr[ (x-3)^2 + (4-0)^2 ]
Finally the other side is sqr[ (x-0)^2 +(0-0)^2 ]=x
Using the property of pythagorean theorem (the triagle is 90 degrees) we find that : x^2=5^2+ {sqr[(x-3)^2 +16]}^2=>

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Re: The two perpendicular lines L and R intersect at the point [#permalink]

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23 Oct 2017, 23:39
honchos wrote:
The two perpendicular lines L and R intersect at the point (3, 4) on the coordinate plane to form a triangle whose other two vertices rest on the x-axis. If one of the other two vertices is located at the origin, what is the x-coordinate of the other vertex?

A. 11/8
B. 33/8
C. 5
D. 20/3
E. 25/3

Another approach to this problem is
By property of right angled triangle BD^2 = AD * DC
4^2 = 3 * DC
DC = 16/3

AC = AD + DC = 3+16/3 = 25/3

Attachments

Triangle.png [ 7.29 KiB | Viewed 319 times ]

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Re: The two perpendicular lines L and R intersect at the point [#permalink]

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23 Oct 2017, 23:43
honchos wrote:
The two perpendicular lines L and R intersect at the point (3, 4) on the coordinate plane to form a triangle whose other two vertices rest on the x-axis. If one of the other two vertices is located at the origin, what is the x-coordinate of the other vertex?

A. 11/8
B. 33/8
C. 5
D. 20/3
E. 25/3

It can be solved by using slope of a line and slope of perpendicular lines approach:
Slope of AB = 4/3
Slope of AC = -3/4 = (4-0)/(3-x)
-9+3x = 16
x = 25/3
Attachments

Triangle.png [ 7.29 KiB | Viewed 316 times ]

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Re: The two perpendicular lines L and R intersect at the point   [#permalink] 23 Oct 2017, 23:43
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