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In the xy-plane, a triangle ABC consists of 3 points A(0,0), B(6,0),

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In the xy-plane, a triangle ABC consists of 3 points A(0,0), B(6,0), and C(0,12). If a point (x,y) is in the triangle ABC, what is the probability that y is smaller than x?

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

* A solution will be posted in two days.
[Reveal] Spoiler: OA

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Re: In the xy-plane, a triangle ABC consists of 3 points A(0,0), B(6,0), [#permalink]

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New post 03 May 2016, 00:09
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we have 2 equations
2x + y = 12
and x = y for the point C
so x = 4 and y = 4
We know that for any point in the triangle OCB, Y<X

so area of OCB is 1/2 x 6 x 4 = 12
Area of OAB is 1/2 x 12x 6 =36

so the probibility is 12/36 = 1/3

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Re: In the xy-plane, a triangle ABC consists of 3 points A(0,0), B(6,0), [#permalink]

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New post 07 May 2016, 08:15
The probability of y<x is same as the area of △OBC/the area of △OBA. Since the lower base=OB=6, we get a/12. Also, if we substitute this into 2x+y=12, we get 2a+12. From this we get a=4. Then, we obtain a/12=4/12=1/3. Hence, the correct answer is B.
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Re: In the xy-plane, a triangle ABC consists of 3 points A(0,0), B(6,0), [#permalink]

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New post 20 Jul 2016, 02:22
MathRevolution wrote:
In the xy-plane, a triangle ABC consists of 3 points A(0,0), B(6,0), and C(0,12). If a point (x,y) is in the triangle ABC, what is the probability that y is smaller than x?

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

* A solution will be posted in two days.

Quote:
Sir,
This attachment that you have in the problem.
Is that information GIVEN or you have derived that drawing based on the information provided in the following question:
In the xy-plane, a triangle ABC consists of 3 points A(0,0), B(6,0), and C(0,12). If a point (x,y) is in the triangle ABC, what is the probability that y is smaller than x?

Thanks,
Yosita

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Re: In the xy-plane, a triangle ABC consists of 3 points A(0,0), B(6,0), [#permalink]

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New post 13 Jan 2017, 06:19
Y would be smaller than x, if the point lies within triangle OBC, i.e., between the lines OB and OC of the given diagram.
C is the intersection AB and OC. To find the coordinates of C we need to solve the equation of AB and OC.

The equation for AB: \(y=ax+b\). Slope of AB is -2, y intersection is 6.

\(∴y=-2x+6=>y+2x=6-------(i)\)

The equation for OC: \(y=x-----(ii)\)

Solving equations (i) and (ii) we get x and y coordinates of point C, x = 4 and y = 4

So, the area of triangle OCB is 4 = 12 and the Area of triangle OAB is 36.

∴ The probability is \(\frac{12}{36}=\frac{1}{3}\)
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Re: In the xy-plane, a triangle ABC consists of 3 points A(0,0), B(6,0), [#permalink]

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In the xy-plane, a triangle ABC consists of 3 points A(0,0), B(6,0), [#permalink]

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New post 26 Aug 2017, 20:57
MathRevolution wrote:
The probability of y<x is same as the area of △OBC/the area of △OBA. Since the lower base=OB=6, we get a/12. Also, if we substitute this into 2x+y=12, we get 2a+12. From this we get a=4. Then, we obtain a/12=4/12=1/3. Hence, the correct answer is B.


It is given y should be less than x , Would this solution account for the integer values where y=x ??

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Siddhuftr wrote:
MathRevolution wrote:
The probability of y<x is same as the area of △OBC/the area of △OBA. Since the lower base=OB=6, we get a/12. Also, if we substitute this into 2x+y=12, we get 2a+12. From this we get a=4. Then, we obtain a/12=4/12=1/3. Hence, the correct answer is B.


It is given y should be less than x , Would this solution account for the integer values where y=x ??


Hello siddhuftr

From the triangle x and y can take non integer values as well , no where it is mentioned about integer values. You have to consider all points in that area.
Hope its clear.


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New post 21 Sep 2017, 01:43
Hi,

attached is how I approached it, and got 1/2 instead of 1/3.
General approach seems to be correct, but I am thinking that the area of smaller triangle shoul be 18, not 12, because the question asks for situation where Y (= height) is less than X (=base), so I drew a triangle (shaded area), where base = 6, height =6.

Where am I wrong with this?? Please enlighten me!! Bunuel

Thank you so much in advance!
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In the xy-plane, a triangle ABC consists of 3 points A(0,0), B(6,0), [#permalink]

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New post 21 Sep 2017, 02:38
HappyQuakka wrote:
Hi,

attached is how I approached it, and got 1/2 instead of 1/3.
General approach seems to be correct, but I am thinking that the area of smaller triangle shoul be 18, not 12, because the question asks for situation where Y (= height) is less than X (=base), so I drew a triangle (shaded area), where base = 6, height =6.

Where am I wrong with this?? Please enlighten me!! Bunuel

Thank you so much in advance!



Your x,y coordinate of point F are wrong. I dont know how you got 6,6 => can you explain how you got 6,6??
and the correct coordinate will be 4,4
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In the xy-plane, a triangle ABC consists of 3 points A(0,0), B(6,0),   [#permalink] 21 Sep 2017, 02:38
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