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

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?

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

* A solution will be posted in two days.

sahilvijay · Accepted Answer

2 methods are posted
Normal way

and shortcut in these questions
hope you will like the approach.

anceer · Answer

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

MathRevolution · Answer

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.

yosita18 · Answer

Mahmud6 · Answer

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}

Siddhuftr · Answer

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

sahilvijay · Answer

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.

Sent from my iPhone using  GMAT Club Forum mobile app

HappyQuakka · Answer

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!

sahilvijay · Answer

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

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards