Problem Solving

Hi ,

Attached is the problem on intercepts. Answer is E.

Can anybody explain how to arrive at this answer.

If a line is parallel to one side of a triangle, that line splits the triangle into proportional sides.
AO= 4
OB=6
Let the parralel line be CD. Let CD have length y. This sets off triangle ACD and trapezoid COBD. We are given that ACD equals COBD, b/c the line divides AOB equally. So:

Area of ACD= Area of COBD
Now let CO =x, which will be the yintercept we are looking for. Since OA is 4 then AC =4-x. We will derive a second equation from the proportionality that results from CD being || to OB

(4-x)/4=y/6

Of course (4-x)/x is also a valid proportion. But here we are taking the whole of AO to be proportional to a part of AO and the parallel line to be proportional to the whole of OB). 4-x and y are from the same triangle. Two unknowns and two equations and we are done.

Area of Triangle ACD= Area of Trapezoid COBD

1/2 (y(4-x)=1/2 (y+6)x ( I'll skip this part)

solve for y and you get

y=(3x/2-x)

From our proportion equation we have

24-6x=4y or
12-3x=2y

Sub y in this we have

12 -3x =2(3x/(2-x)
12(2-x)-3x(2-x)=6x
24-12x-6x+3x2=6x
3x2-24x+24=0
x2-8x+8=0
(8+-(64-32)^.5)/2
4+-2(2)^.5
4+2(2)^.5 is invalid since it is greater than AO.
4-2(2)^.5 is your desired E.

The explanation is long but the problem is not difficult just a too much calculation. with speed and no errors can be solved in about 3.5. I am not certain there is a shorter route around these two equations but I would think about one or hope someone can.

Hope this helps. This is a good problem to refresh on similar triangles which the GMAT folks love.

Thanks a lot.

I really dont think you need to solve it.. coz doing so you are definitely going to take more than 3 to 4 mins.. under the exam pressure..

Simple way is to use process of elimination.. or POE..with some basic calculations
We have 5 choices.. just by looking at the choices I can remove a, c and d.. why? well, here's the explanation:

we know root2 = 1.414 and root3 = 1.732

so a) = 3.414, c) = 3.732 and d) = 5.414
all these values are either very close to 4 or more than 4.. by looking at the figure its pretty easy to make out that the area of the trapezoid, if it has to be half of the original triangle, then the y intercept has to be a small number..

So now I am left with b) and e)

As per the question, if the y-intercept is denoted as y then the point where line k intersects the line AB will be (x,y)
Area of trapezoid = 1/2(x+6)(y) = 6 (half of the area of the triangle)

=> x = 12/y - 6

just substitute b and e, one by one.. b = 2.6 (approx) and e = 1.2 (approx)

for b we get x as negative.. which is not possible.. so the only choice left is E

Hope this is helpful..

dumb.doofus wrote:I really dont think you need to solve it.. coz doing so you are definitely going to take more than 3 to 4 mins.. under the exam pressure..

Simple way is to use process of elimination.. or POE..with some basic calculations
We have 5 choices.. just by looking at the choices I can remove a, c and d.. why? well, here's the explanation:

we know root2 = 1.414 and root3 = 1.732

so a) = 3.414, c) = 3.732 and d) = 5.414
all these values are either very close to 4 or more than 4.. by looking at the figure its pretty easy to make out that the area of the trapezoid, if it has to be half of the original triangle, then the y intercept has to be a small number..

So now I am left with b) and e)

As per the question, if the y-intercept is denoted as y then the point where line k intersects the line AB will be (x,y)
Area of trapezoid = 1/2(x+6)(y) = 6 (half of the area of the triangle)

=> x = 12/y - 6

just substitute b and e, one by one.. b = 2.6 (approx) and e = 1.2 (approx)

for b we get x as negative.. which is not possible.. so the only choice left is E

Hope this is helpful..

Nice assumptions, but it can be risky since these figures are not drawn to scale.Either way it is long. You almost never look at at a geometry problem and go I am using POE. Your first instinct is to figure out the problem . You will probably use 2 minutes before you realize you have no clue. That 2 minutes was spent on a different objective or strategy so it may have been wasted. Almost all geometry problems have the temptation that you think you can reason it out. Similar triangle, sum of angles = 180 in triangle, etc. By the time you figure it out by POE, you have lost another 2. You will still end end up around the 4 min mark on such a question if you ever run into it. The key is either way, You had better get it right or else you were better off guessing and moving on.

dtweah wrote:

dumb.doofus wrote:I really dont think you need to solve it.. coz doing so you are definitely going to take more than 3 to 4 mins.. under the exam pressure..

Simple way is to use process of elimination.. or POE..with some basic calculations
We have 5 choices.. just by looking at the choices I can remove a, c and d.. why? well, here's the explanation:

we know root2 = 1.414 and root3 = 1.732

so a) = 3.414, c) = 3.732 and d) = 5.414
all these values are either very close to 4 or more than 4.. by looking at the figure its pretty easy to make out that the area of the trapezoid, if it has to be half of the original triangle, then the y intercept has to be a small number..

So now I am left with b) and e)

As per the question, if the y-intercept is denoted as y then the point where line k intersects the line AB will be (x,y)
Area of trapezoid = 1/2(x+6)(y) = 6 (half of the area of the triangle)

=> x = 12/y - 6

just substitute b and e, one by one.. b = 2.6 (approx) and e = 1.2 (approx)

for b we get x as negative.. which is not possible.. so the only choice left is E

Hope this is helpful..

Nice assumptions, but it can be risky since these figures are not drawn to scale.Either way it is long. You almost never look at at a geometry problem and go I am using POE. Your first instinct is to figure out the problem . You will probably use 2 minutes before you realize you have no clue. That 2 minutes was spent on a different objective or strategy so it may have been wasted. Almost all geometry problems have the temptation that you think you can reason it out. Similar triangle, sum of angles = 180 in triangle, etc. By the time you figure it out by POE, you have lost another 2. You will still end end up around the 4 min mark on such a question if you ever run into it. The key is either way, You had better get it right or else you were better off guessing and moving on.

hey buddy.. I understand your point.. but here is why I took this approach...

1. This question doesnt require you to think that what is drawn may not be of scale.. actually to what scale it is drawn is immaterial here.. for the basic fact that I know that the area of trapezium here is half of the area of the triangle.. and the sides are 4 and 6 of that triangle.. since the area of a trapezium involves adding the parallel sides, it is quite evident that the length between those parallel sides should be quite small, as the parallel sides are pretty lengthy in the context of the problem..

2. In terms of deriving at the answer in less than 2 mins.. well, yes I did.. those assumptions are very logical.. and not a shot in the dark..

But then in the end... I have to say like I always do.. the above solution is the simplest solution in "just my opinion" Nothing more than that..

No offense to you or to anyone..

dumb.doofus wrote:

dtweah wrote:

dumb.doofus wrote:I really dont think you need to solve it.. coz doing so you are definitely going to take more than 3 to 4 mins.. under the exam pressure..

Simple way is to use process of elimination.. or POE..with some basic calculations
We have 5 choices.. just by looking at the choices I can remove a, c and d.. why? well, here's the explanation:

we know root2 = 1.414 and root3 = 1.732

so a) = 3.414, c) = 3.732 and d) = 5.414
all these values are either very close to 4 or more than 4.. by looking at the figure its pretty easy to make out that the area of the trapezoid, if it has to be half of the original triangle, then the y intercept has to be a small number..

So now I am left with b) and e)

As per the question, if the y-intercept is denoted as y then the point where line k intersects the line AB will be (x,y)
Area of trapezoid = 1/2(x+6)(y) = 6 (half of the area of the triangle)

=> x = 12/y - 6

just substitute b and e, one by one.. b = 2.6 (approx) and e = 1.2 (approx)

for b we get x as negative.. which is not possible.. so the only choice left is E

Hope this is helpful..

Nice assumptions, but it can be risky since these figures are not drawn to scale.Either way it is long. You almost never look at at a geometry problem and go I am using POE. Your first instinct is to figure out the problem . You will probably use 2 minutes before you realize you have no clue. That 2 minutes was spent on a different objective or strategy so it may have been wasted. Almost all geometry problems have the temptation that you think you can reason it out. Similar triangle, sum of angles = 180 in triangle, etc. By the time you figure it out by POE, you have lost another 2. You will still end end up around the 4 min mark on such a question if you ever run into it. The key is either way, You had better get it right or else you were better off guessing and moving on.

hey buddy.. I understand your point.. but here is why I took this approach...

1. This question doesnt require you to think that what is drawn may not be of scale.. actually to what scale it is drawn is immaterial here.. for the basic fact that I know that the area of trapezium here is half of the area of the triangle.. and the sides are 4 and 6 of that triangle.. since the area of a trapezium involves adding the parallel sides, it is quite evident that the length between those parallel sides should be quite small, as the parallel sides are pretty lengthy in the context of the problem..

2. In terms of deriving at the answer in less than 2 mins.. well, yes I did.. those assumptions are very logical.. and not a shot in the dark..

But then in the end... I have to say like I always do.. the above solution is the simplest solution in "just my opinion" Nothing more than that..

No offense to you or to anyone..

No dumb I am not offended. If I were I wouldn't say nice assumptions. (laugh). I am just looking at the dynamics of a difficult question poping at you and the what a given program of attack you choose. To me POE is a strategy of second resorts and many problems are not amenable to POE. This problem may have been but the next one may not be. I will be hard pressed to believe that you get a geometry question and go straight for POE. I probably would not recommend that.