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Manager  Joined: 19 Apr 2011
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Point (x,y) is a point within the triangle. What is the  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 45% (02:30) correct 55% (02:16) wrong based on 355 sessions

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Point (x,y) is a point within the triangle. What is the probability that y<x?

A. 1/4
B. 1/3
C. 1/6
D. 1/2
E. 1/5

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+1 if you like my explanation .Thanks Originally posted by saikarthikreddy on 23 Sep 2012, 05:16.
Last edited by Bunuel on 24 May 2013, 01:52, edited 1 time in total.
Edited the OA.
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Re: Point (x,y) is a point within the triangle.  [#permalink]

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saikarthikreddy wrote:
Point (x,y) is a point within the triangle. What is the probability that y<x?

a. 1/4
b. 1/8
c. 1/6
d. 1/2
e. 1/5

1. Use the x=y boundary line. The region of the triangle below this line contains points x > y.
2. Get the line that of the triangle.

$$m = \frac{10 - 0}{0-5} = -2$$

$$y = -2x + b$$
$$10 = -2(0) + b$$
$$b=10$$

$$Line: y = -2x + 10$$

3. Get the point of intersection of y=x and y=-2x+10.
$$x = -2x + 10$$
$$3x = 10$$
$$x = 10/3$$

4. Get the area of smaller triangle: $$=\frac{10}{3}*\frac{1}{2}*5=\frac{25}{3}$$
5. Get the area of the larger triangle: $$10*5*\frac{1}{2} = 25$$
6. $$\frac{smallerArea}{largerArea}=\frac{25}{3}*\frac{1}{25}=\frac{1}{3}$$

##### General Discussion
Intern  Joined: 15 Sep 2012
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GMAT Date: 10-18-2012
Re: Point (x,y) is a point within the triangle.  [#permalink]

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In the given triangle y<x, when value of x => 3.35(approx.)

So, we we form a triangle by taking this point(3.35,3.3) in the given triangle. The solution to above question will be the area of the smaller triangle divided by the larger one.

Area of smaller triangle(2.85)/Area of larger triangle(25)

I am not sure of my approach. So experts, please can you advise if my approach is correct.
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Re: Point (x,y) is a point within the triangle.  [#permalink]

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Prob(y<x) = $$\frac{Area(small Triangle)}{Area(large Triangle)}$$

Area(large Triangle) = 25

Area(small Triangle) = ?

The small triangle would be made up of 3 points:
1. the origin
2. (5,0),
3. a point on the hypotenuse where y=x

To figure out this point we build the equation for the hypotenuse, y = 10 - 2x, and calculate the intersection with y = x, solve the system to find that x = $$\frac{10}{3}$$ = y

Area(small Triangle) = $$\frac{1}{2}(5 * \frac{10}{3} ) = \frac{50}{6}$$

$$Prob (y<x) = \frac{50}{6}/25 = \frac{1}{3}$$

Any chance 1/8 could've been 1/3?
Intern  Joined: 09 Sep 2012
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Location: United States
Re: Point (x,y) is a point within the triangle.  [#permalink]

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leigimon wrote:

Prob(y<x) = $$\frac{Area(small Triangle)}{Area(large Triangle)}$$

Area(large Triangle) = 25

Area(small Triangle) = ?

The small triangle would be made up of 3 points:
1. the origin
2. (5,0),
3. a point on the hypotenuse where y=x

To figure out this point we build the equation for the hypotenuse, y = 10 - 2x, and calculate the intersection with y = x, solve the system to find that x = $$\frac{10}{3}$$ = y

Area(small Triangle) = $$\frac{1}{2}(5 * \frac{10}{3} ) = \frac{50}{6}$$

$$Prob (y<x) = \frac{50}{6}/25 = \frac{1}{3}$$

Any chance 1/8 could've been 1/3?

Hi guys.. Can you please explain how did you find the equation of hypotenuse.. y=mx+c >> m=-2 , why c= 10 ?
Intern  Joined: 24 Sep 2012
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Re: Point (x,y) is a point within the triangle.  [#permalink]

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154238 wrote:
Hi guys.. Can you please explain how did you find the equation of hypotenuse.. y=mx+c >> m=-2 , why c= 10 ?

c usually refers to the y-intercept in that form. From the drawing we can see that the y-intercept (where x = 0) is 10.
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Re: Point (x,y) is a point within the triangle.  [#permalink]

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leigimon wrote:
154238 wrote:
Hi guys.. Can you please explain how did you find the equation of hypotenuse.. y=mx+c >> m=-2 , why c= 10 ?

c usually refers to the y-intercept in that form. From the drawing we can see that the y-intercept (where x = 0) is 10.

Thanks a lot buddy !!
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Re: Point (x,y) is a point within the triangle.  [#permalink]

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1
1
154238 wrote:
leigimon wrote:

Prob(y<x) = $$\frac{Area(small Triangle)}{Area(large Triangle)}$$

Area(large Triangle) = 25

Area(small Triangle) = ?

The small triangle would be made up of 3 points:
1. the origin
2. (5,0),
3. a point on the hypotenuse where y=x

To figure out this point we build the equation for the hypotenuse, y = 10 - 2x, and calculate the intersection with y = x, solve the system to find that x = $$\frac{10}{3}$$ = y

Area(small Triangle) = $$\frac{1}{2}(5 * \frac{10}{3} ) = \frac{50}{6}$$

$$Prob (y<x) = \frac{50}{6}/25 = \frac{1}{3}$$

Any chance 1/8 could've been 1/3?

Hi guys.. Can you please explain how did you find the equation of hypotenuse.. y=mx+c >> m=-2 , why c= 10 ?

When m and c are not 0, the line is not horizontal and will not pass through the origin. Then both the x and the y intercept will be non-zero.
The y intercept is the value of y for x = 0, which, for the equation y = mx + c, is c.
The x intercept is the value of x for y = 0, which is -c/m.
The given equation y = mx + c can be rewritten as -mx + y = c,
or $$\frac{x}{-c/m}+\frac{y}{c}=1$$. You can see that the denominator of x is exactly the x intercept and the denominator of y is the y intercept.

Each line which doesn't go through the origin, has its equation as $$\frac{x}{x_i}+\frac{y}{y_i}=1$$ , where $$x_i$$ and $$y_i$$ are the x and the y intercept, respectively.

In our case, we could have written directly the equation of the hypotenuse as $$\frac{x}{5}+\frac{y}{10}=1$$ which we can rearrange and get $$y=-2x+10.$$

So, next time, if you have the two intercepts, for example you know that the line goes through the points (-3,0) and (0,4), you can immediately write the equation of the line as $$\frac{x}{-3}+\frac{y}{4}=1$$ rearrange as you wish...
I mean you can save the time of finding the slope and write the standard equation of a line... Not that it is such a saving, but anyway, it is a nice mathematical property _________________
PhD in Applied Mathematics
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Point (x,y) is a point within the triangle.  [#permalink]

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Why are u guys calculating the triangle area.. with 1/2 *b*h ???

Its not a right triangle right??
Senior Manager  Joined: 13 Aug 2012
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Re: Point (x,y) is a point within the triangle.  [#permalink]

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Shrek89 wrote:
Why are u guys calculating the triangle area.. with 1/2 *b*h ???

Its not a right triangle right??

We have to get the portion of the triangle (0,0), (10,0) and (5,0) with x>y. A boundary line x=y will divide this triangle to two portions: a portion with x>y and a portion with x<y. Now you have to get the desired portion which is the smaller triangle below the x=y boundary but still within the main triangle. That's why we are calculating two areas: $$\frac{desired portion}{main triangle}$$
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Re: Point (x,y) is a point within the triangle.  [#permalink]

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Ya...but the desired area cant be calculated as we do for right triangle as it is not a right triangle....

Jus look into that once..
Senior Manager  Joined: 13 Aug 2012
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Re: Point (x,y) is a point within the triangle.  [#permalink]

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That is a good question. But you can still calculate even if it is not a right triangle.

The smaller triangle is formed by the coordinates (0,0), (10/3,10/3) and (5,0). This is not a right triangle but you are given its height through its (10/3,10/3) coordinate. Then your base is 5 which is equal to the base of your main triangle.

Hope it helps.

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Re: Point (x,y) is a point within the triangle.  [#permalink]

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Thanks mbaiseasy ....I am being dumb lately :D ....Thanks for repeated explanations Intern  Joined: 15 May 2012
Posts: 26
Re: Point (x,y) is a point within the triangle.  [#permalink]

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leigimon wrote:

Prob(y<x) = $$\frac{Area(small Triangle)}{Area(large Triangle)}$$

Area(large Triangle) = 25

Area(small Triangle) = ?

The small triangle would be made up of 3 points:
1. the origin
2. (5,0),
3. a point on the hypotenuse where y=x

To figure out this point we build the equation for the hypotenuse, y = 10 - 2x, and calculate the intersection with y = x, solve the system to find that x = $$\frac{10}{3}$$ = y

Area(small Triangle) = $$\frac{1}{2}(5 * \frac{10}{3} ) = \frac{50}{6}$$

$$Prob (y<x) = \frac{50}{6}/25 = \frac{1}{3}$$

Any chance 1/8 could've been 1/3?

Hi, What if the question is "what is the probability that y>x?"
Then, is the answer 2/3 correct? One way to calculate is 1-1/3. But, if the question directly asks the prob. of y>x, by using your method I got 2/3. I am sure this is right, but please confirm.
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Posts: 60480
Re: Point (x,y) is a point within the triangle.  [#permalink]

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sharmila79 wrote:
leigimon wrote:

Prob(y<x) = $$\frac{Area(small Triangle)}{Area(large Triangle)}$$

Area(large Triangle) = 25

Area(small Triangle) = ?

The small triangle would be made up of 3 points:
1. the origin
2. (5,0),
3. a point on the hypotenuse where y=x

To figure out this point we build the equation for the hypotenuse, y = 10 - 2x, and calculate the intersection with y = x, solve the system to find that x = $$\frac{10}{3}$$ = y

Area(small Triangle) = $$\frac{1}{2}(5 * \frac{10}{3} ) = \frac{50}{6}$$

$$Prob (y<x) = \frac{50}{6}/25 = \frac{1}{3}$$

Any chance 1/8 could've been 1/3?

Hi, What if the question is "what is the probability that y>x?"
Then, is the answer 2/3 correct? One way to calculate is 1-1/3. But, if the question directly asks the prob. of y>x, by using your method I got 2/3. I am sure this is right, but please confirm.

Yes, you are right.

Similar questions to practice:
in-the-xy-plane-a-triangle-has-vertexes-0-0-4-0-and-88395.html
in-the-coordinate-plane-rectangular-region-r-has-vertices-a-104869.html
set-t-consists-of-all-points-x-y-such-that-x-2-y-2-1-if-15626.html

Hope it helps.
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Re: Point (x,y) is a point within the triangle.  [#permalink]

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There is a formula for area of triangle if lengths of two sides and the included angle is known. Area = 1/2 * a * b * sinC.

If we draw x=y line, we get 2 triangles.

Answer = area of upper triangle/total area.

If we take the length of the x=y line as p, then
total area = upper tri + lower tri
= (1/2 * 10 * p * sin45) + (1/2 * p * 5 * sin 45)

Ignoring 1/2 , p, sin 45 in calculation.

total area = 15.
upper area = 10.

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Re: Point (x,y) is a point within the triangle. What is the  [#permalink]

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saikarthikreddy wrote:
Attachment:
The attachment triangle.jpg is no longer available
Point (x,y) is a point within the triangle. What is the probability that y<x?

A. 1/4
B. 1/3
C. 1/6
D. 1/2
E. 1/5

here is my answer. Hope it's clear.
Attachments 1.jpg [ 47.14 KiB | Viewed 10061 times ]

Intern  Joined: 21 Apr 2014
Posts: 38
Re: Point (x,y) is a point within the triangle. What is the  [#permalink]

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1
so one way to think of this is the area of the triangle where x>y over the total area of the triangle.

The ares of the triangle can be discovered by first finding where x=y and the hypotenuse meet. We know two points on the hypotenuse (0,10) and (5,0), so we know the y intercept is 10 and the slope is (-10/5) or -2. y=-2x+10

That meets y=x when x=-2x+10 or when 3x=10, so x=10/3 and y =10/3

we now know the height of the triangle (10/3) and the base of the triangle 5, so the total area is 50/6 or 25/3

That divided by 25 (the total area of the triangle) equals 1/3 (B)
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Re: Point (x,y) is a point within the triangle. What is the  [#permalink]

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One question ... is this approach correct?

if slope of the hyp --> y=2x
and slope the bisect --> y=x

For each x I have 2 y --> so I can see this as a ratio question:

x : y : total
1: 2 : 3

So the p(x>y) = 1/3
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