D01-34

Question

If vertices of a triangle are A (5, 0), B (x, y) and C (25, 0), what is the area of the triangle?

(1)  |x| = y = 10 
 
(2)  x = |y| = 10

Bunuel · Answer

Official Solution:

If vertices of a triangle are A (5, 0), B (x, y) and C (25, 0), what is the area of the triangle? 
 
(1)  |x| = y = 10 
 
The above statement means that  y  is 10, but  x  can be either -10 or 10. Thus, the third vertex B can be either at (-10, 10) or (10, 10). Notice, however, that the area will be the same in either case. The base of the triangle, the red segment in the image below, will have a length of 20, and the height from B will be 10. Therefore, the area of the triangle will be  \frac{1}{2}*20*10 = 100 . Sufficient.
 
 https://gmatclub.com/gmat-practice-tests/resources/import/img/D01-34-1+.png 
 
(2)  x = |y| = 10 
 
The above statement means that  x  is 10, but  y  can be either -10 or 10. Thus, the third vertex B can be either at (10, 10) or (10, -10). Notice, however, that the area will be the same in either case. The base of the triangle, the red segment in the image below, will have a length of 20, and the height from B will be 10. Therefore, the area of the triangle will be  \frac{1}{2}*20*10 = 100 . Sufficient.
 
 https://gmatclub.com/gmat-practice-tests/resources/import/img/D01-34-2.png

Answer: D

sid200207 · Answer

Can you please explain statement one in more detail i.e. how area is same .

arunspanda · Answer

If vertices of a triangle are A (5, 0), B (x, y) and C (25, 0), what is the area of the triangle?

(1) |x| = y = 10

(2) x = |y| = 10

Given that vertices of a triangle are A (5, 0) and C (25, 0) =>Two vertices of the triangle are on x-axis =>Base = 20.
Area of a triangle = 1/2*(base)*(height)
Since, base is given, area of the triangle can be found if any of the following statement uniquely provide  height  of the triangle.

Statement 1:  
|x| = y = 10 
Or, x point lies on the line y=10 => height of the triangle=10 => sufficient..... (B)(C)(E) 
 Statement 2: 
 x = |y| = 10
Or, y point lies on the line x=10 => height of the triangle =10 => sufficient..... (A)  (D)

Answer: (D)

Bunuel · Answer

Look at the diagram below: Untitled.png Notice that no matter where point B is (blue or green dot) the area of the triangle would be the same, because the base (red segment) and the height would be the same.

Thoughtosphere · Answer

Dear  Bunuel ,

Please correct me, if my reasoning for the above question is wrong.

The two specified co-ordinates will be on X-Axis. 
(1) Y = 10 and the co-ordinate of x can be either 10 or -10, but in both the cases, X will lie on a straight line that is parallel to X Axis, thus we can use the Triangle area concept that if two points of a triangle lie on a line then the all the triangles formed with the third point on the parallel line will have the same area.

(2) The same concept can be used here. Though in this case, it is visible that the two triangles formed will be mirror image of each other, thus even in this case the area can be known.

Thus Answer should be D

23a2012 · Answer

Dear Bunuel, that is the same what i am draw and that is what i am ask for. In the green triangle I need to understand which side is the height and how it equal to 10.I really confused

Bunuel · Answer

I think that you really need to brush up fundamentals on geometry.

prabsahi · Answer

I think this is a  high-quality  question and I agree with explanation. loved the concept!!

prabsahi · Answer

I think this is a  high-quality  question. loved it.

psaikrishna90 · Answer

I think this is a  high-quality  question and I agree with explanation.

MWithrock · Answer

The graph did everything. I am so mad I missed this question now.

patto · Answer

I think this is a  high-quality  question and I agree with explanation. I don't think this is a 700 question..

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

kop18 · Answer

Very useful question that tests your knowledge of height in an acute and obtuse triangle using a graph.

The 2nd statement according to me is quite self explanatory however for the first statement - 
The easiest way to approach this question would be to use to think of the triangles in terms of obtuse and acute triangles. 
The 2 triangles we obtain are case 1 => x = -10 (obtuse) and case 2 => x=10 (acute). For an acute triangle the height is always inside the triangle whereas for an obtuse triangle it always outside the triangle. The moment we drop the height for the triangles we observe that in both the cases the height is 10 and the base (the red segment on @bunuels explanation is the base for both the triangles). Since the base remains the same and so does the height we get the same area for the both the points of x i.e, -10 and 10.

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