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M04-12

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Bunuel
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M04-12 [#permalink] New post   16 Sep 2014, 00:22
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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.
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Re: M04-12 [#permalink] New post   27 Apr 2015, 01:44
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ishitathukral wrote:
first option says AB =6
doesnt it imply that the sides are 6,8,10?


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.
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Re: M04-12 [#permalink] New post   31 Jan 2015, 06:13
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chieffarmer wrote:
Bunuel wrote:
Official Solution:


Since all points are distinct and BD is a height then B must be a right angle and AC must be a hypotenuse (so BD is a height from right angle B to the hypotenuse AC). Question thus asks about the product of non-hypotenuse sides AB and BC.

(1) AB = 6. Clearly insufficient.

(2) The product of the non-hypotenuse sides is equal to 24 \(\rightarrow\) directly gives us the value of AB*BC. Sufficient.


Answer: B


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.


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).
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Re M04-12 [#permalink] New post   16 Sep 2014, 00:22
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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
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Re: M04-12 [#permalink] New post   30 Jan 2015, 14:55
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Bunuel wrote:
Official Solution:


Since all points are distinct and BD is a height then B must be a right angle and AC must be a hypotenuse (so BD is a height from right angle B to the hypotenuse AC). Question thus asks about the product of non-hypotenuse sides AB and BC.

(1) AB = 6. Clearly insufficient.

(2) The product of the non-hypotenuse sides is equal to 24 \(\rightarrow\) directly gives us the value of AB*BC. Sufficient.


Answer: B


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.
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Re: M04-12 [#permalink] New post   26 Apr 2015, 18:31
first option says AB =6
doesnt it imply that the sides are 6,8,10?
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Re: M04-12 [#permalink] New post   08 Jul 2015, 09:42
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.
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Re: M04-12 [#permalink] New post   21 Jul 2015, 04:32
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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.
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Re: M04-12 [#permalink] New post   22 Jul 2015, 01:16
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CountClaud wrote:
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.


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.
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Re: M04-12 [#permalink] New post   12 Jan 2016, 07:50
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buffaloboy wrote:
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.


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.
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Re: M04-12 [#permalink] New post   29 Apr 2016, 13:35
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Bunuel wrote:
CountClaud wrote:
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.


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.



How about this figure? It doesn't say 'D' is a point on the triangle, just says D is a point on the plane...
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Re: M04-12 [#permalink] New post   02 May 2016, 04:10
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nk18967 wrote:
Bunuel wrote:
CountClaud wrote:
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.


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.



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.
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Re: M04-12 [#permalink] New post   02 May 2016, 04:26
How about this figure? It doesn't say 'D' is a point on the triangle, just says D is a point on the plane...[/quote]

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.[/quote]

-----

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!
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Re: M04-12 [#permalink] New post   02 May 2016, 04:30
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nk18967 wrote:
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.[/quote]

-----

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![/quote]

It's not necessary to be inside a triangle:
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Re: M04-12 [#permalink] New post   24 May 2016, 18:57
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.
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Re: M04-12 [#permalink] New post   25 May 2016, 08:43
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DavidFox wrote:
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.


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.
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Re: M04-12 [#permalink] New post   04 Dec 2017, 22:32
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 ?
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Re: M04-12 [#permalink] New post   04 Dec 2017, 22:48
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chetanb12 wrote:
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 ?


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?
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Re: M04-12 [#permalink] New post   01 Jan 2018, 06:28
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
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Re: M04-12 [#permalink] New post   01 Jan 2018, 06:31
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sandysilva wrote:
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


(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.
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