M04-12

Question

On a plane, there are four distinct points A, B, C, and D. If triangle ABC is a right-angled triangle and BD is one of its heights, what is the product of the length of AB and BC?

(1)  AB = 6 
 
(2) The two sides of triangle ABC that are not the hypotenuse have a product of 24.

Bunuel · Accepted Answer

No. Knowing that one side of a right triangle is 6 DOES NOT mean that the sides of the right triangle necessarily must be in the ratio of Pythagorean triple - 6:8:10. Or in other words: if 6^2+y^2=z^2 DOES NOT mean that y=8 and z=10 . Certainly this is one of the possibilities but definitely not the only one. In fact 6^2+y^2=z^2 has infinitely many solutions for y and z and only one of them is y=8 and z=10 . For example: y=1 and z=\sqrt{37} or y=2 and z=\sqrt{40} ... For more on this trap check the following questions: what-is-the-area-of-parallelogram-abcd-111927.html the-circular-base-of-an-above-ground-swimming-pool-lies-in-a-167645.html figure-abcd-is-a-rectangle-with-sides-of-length-x-centimete-48899.html in-right-triangle-abc-bc-is-the-hypotenuse-if-bc-is-13-and-163591.html m22-73309-20.html points-a-b-and-c-lie-on-a-circle-of-radius-1-what-is-the-84423.html if-vertices-of-a-triangle-have-coordinates-2-2-3-2-and-82159-20.html if-p-is-the-perimeter-of-rectangle-q-what-is-the-value-of-p-135832.html if-the-diagonal-of-rectangle-z-is-d-and-the-perimeter-of-104205.html what-is-the-area-of-rectangular-region-r-105414.html what-is-the-perimeter-of-rectangle-r-96381.html pythagorean-triples-131161.html given-that-abcd-is-a-rectangle-is-the-area-of-triangle-abe-127051.html m13-q5-69732-20.html#p1176059 m20-07-triangle-inside-a-circle-71559.html what-is-the-perimeter-of-rectangle-r-96381.html what-is-the-area-of-rectangular-region-r-166186.html if-distinct-points-a-b-c-and-d-form-a-right-triangle-abc-129328.html Hope this helps.

Bunuel · Answer

BD is a height means that B is a right angle and AC is a hypotenuse (so BD is a height from right angle B to the hypotenuse AC).
 https://gmatclub.com/forum/download/file.php?id=17467

Bunuel · Answer

Official Solution:

On a plane, there are four distinct points A, B, C, and D. If triangle ABC is a right-angled triangle and BD is one of its heights, what is the product of the length of AB and BC?  
 
Since the points A, B, C, and D are distinct and BD is a height of the right-angled triangle ABC, we know that point B must be the right angle and AC must be the hypotenuse. This means that BD is the height from right angle B to hypotenuse AC. So, the problem is asking us to find the product of the two non-hypotenuse sides AB and BC.
 
(1) AB = 6. Clearly insufficient.
 
(2) The two sides of triangle ABC that are not the hypotenuse have a product of 24.
 
As discussed above, these two sides are AB and BC. Therefore, we can directly find the value of AB*BC to be 24. Sufficient.

Answer: B

chieffarmer · Answer

How do you know that B is a right angle? Couldn't we also have BC or BA as a hypotenuse with BD still indicating the height of the triangle?

Thx in advance for your help.

ishitathukral · Answer

first option says AB =6
doesnt it imply that the sides are 6,8,10?

buffaloboy · Answer

I think this is a  poor-quality  question and the explanation isn't clear enough, please elaborate. I am surprised if A,B, C, and D are distinct points and ABC is right triangle, right angled at B, then how come A , and D are not the same point. AB is the height . I am damn confused.

CountClaud · Answer

Hi Bunuel,

Can you please explain why the triangle cannot have A or C as the right angle, and the height BD drawn outside of the triangle? If you flip the triangle upside down, it looks to me that you can have a height outside of the triangle, which would make the location of the right angle ambiguous.

Bunuel · Answer

If A is a right angle, so if BA is perpendicular to CA, then the height from B to CA will be BA, making A and D to coincide, which is not possible since we are told that A, B, C, and D are distinct points on a plane. The same if C is a right angle.

saran3129 · Answer

Each triangle can have three different bases and perpendicular to each base, three different heights. In the given description of triangle ABC, angle B being right angle, if you take BA or BC as height, then D coincides with A or C. But the question also says all four points are distinct. Hence we need to take AC as base and BD as height. Thats why this problem is in 700-800 difficulty level. Hope this helps.

NYCBS19 · Answer

How about this figure? It doesn't say 'D' is a point on the triangle, just says D is a point on the plane...

Bunuel · Answer

The figure is NOT right. We are given that BD is a height of the triangle. The height is a  perpendicular  dropped from one of the vertices to the opposite side.

NYCBS19 · Answer

How about this figure? It doesn't say 'D' is a point on the triangle, just says D is a point on the plane...

The figure is NOT right. We are given that BD is a height of the triangle. The height is a  perpendicular  dropped from one of the vertices to the opposite side.

-----

Hmm...I guess my concept of height wasn't clear. So, height of a triangle is ALWAYS the altitude of the triangle, inside the triangle.. and that altitude changes depending on which side is the 'base'.
Thanks!

Bunuel · Answer

The figure is NOT right. We are given that BD is a height of the triangle. The height is a perpendicular dropped from one of the vertices to the opposite side. ----- Hmm...I guess my concept of height wasn't clear. So, height of a triangle is ALWAYS the altitude of the triangle, inside the triangle.. and that altitude changes depending on which side is the 'base'. Thanks! It's not necessary to be inside a triangle: alt2.gif

DavidFox · Answer

I think this is a  poor-quality  question and I don't agree with the explanation. Solution is incorrect. Answer should be (E). There is no way to prove that AB or BC are both legs, or alternatively that one is a leg and the other a hypotenuse. The height BD can lie outside the triangle contrary to the discussion posted in the forum.

Bunuel · Answer

That's not correct. The height in a right triangle is either one of the legs or the perpendicular from right angle to the hypotenuse. Thus the height of a right triangle cannot lie outside the triangle. Moreover, since A, B, C, and D are distinct points then BD cannot be the height from non-right angle because in this case it would coincide with one of the legs.

chetanb12 · Answer

In the second statement, it says the product of 2 non hypotenuse sides is 24. Then in that case, it can have multiple answers such as 2*12, 3*8, 4*6. With help of statement 1, we can say that AB*AC is 6*4 only. Hence, my doubt is, why cant the answer be C ?

Bunuel · Answer

The question asks to find the value of AB*BC. (2) says that AB*BC = 24. We have our answer! Does it matter whether it's 1*24, 1/2*48, 500*24/500, or anything else?

sandysilva · Answer

Hi Bunuel.

Statement 2 says that the product of non hypotenuse sides is 24. Why are we not considering BD here?

Can among AB, BC or BD, any of the two sides be non hypotenuse side ?

Happy new Year.

Regards

Bunuel · Answer

(2) says: The product of the non-hypotenuse sides of triangle ABC is equal to 24. In triangle ABC, AC is hypotenuse, so non-hypotenuse sides are AB and BC. I think this is shown/explained on previous two pages several times.

M04-12

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