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605-655 (Medium)|   Geometry|      
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A, B, C, and D are distinct points on a plane. If triangle A 02 Feb 2015, 01:52
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23a2012
qlx
enigma123
If distinct points A, B, C, and D form a right triangle ABC with a height BD, what is the value of AB times BC ?

(1) AB = 6.
(2) The product of the non-hypotenuse sides is equal to 24.

I am struggling to understand the solution given in GMATCLub Test. Can you please explain how statement 2 is sufficient to answer this question?


Attachment:
TRIANGLE 1.jpg
If the triangle is as shown in the attachment above then in statement 2 when it says the product of the non hypotenuse sides is 24
then we could also take in ∆ ABD hypotenuse is AB and non hypotenuse sides as AD *BD

also in ∆ BDC hypotenuse is BC and non hypotenuse sides are BD*DC

So statement 2 seems a bit ambiguous.

There are actually 3 right triangles in points ABCD , ∆ ABC, ∆ ABD, ∆ BDC so when they say product of non hypotenuse side
is 24 , we could choose any right triangle among these 3 and take 2 of its non hypotenuse sides .

Why are we assuming that the question is talking of ∆ ABC when it says product of non hypotenuse sides
is 24.

If question had said product of non hypotenuse sides of largest triangle or product of non hypotenuse sides of triangle ABC then 2 could surely be enough.

Please correct me if I am wrong.

Dear Bunuel,I agree with glx and I asked myself the same question above. Why we

assuming that the question is talking about ∆ ABC while the statement 2 did not say that the product of non

hypotenuse sides of largest triangle is 24 ? .
Show more

Please check here: a-b-c-and-d-are-distinct-points-on-a-plane-if-triangle-a-129328.html#p1350829
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A, B, C, and D are distinct points on a plane. If triangle A 06 Mar 2015, 11:09
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I think most users are confused on this question because they think why B must be right angle. However, we need to understand that every right triangle has 3 heights. One is from the right angle to the hypotenuse, the others are small legs. And in this case we are given that D is a distinct point than other points. So BD cannot be a small leg. Am I right Bunuel?
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A, B, C, and D are distinct points on a plane. If triangle A 11 May 2015, 21:42
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Ergenekon
I think most users are confused on this question because they think why B must be right angle. However, we need to understand that every right triangle has 3 heights. One is from the right angle to the hypotenuse, the others are small legs. And in this case we are given that D is a distinct point than other points. So BD cannot be a small leg. Am I right Bunuel?
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Consider the below diagram for a normal triangle and for a right angled triangle. For a normal triangle there are three altitudes/heights, one each from the vertex to their opposite bases.

For a right angled triangle, two of its legs form the altitudes to the opposite bases. So BC is an altitiude from the vertex C to base AB and AB is an altitude from the vertex A to base BC. That leaves us with only one altitude which is not a side of the right angled triangle i.e. BD which is an altitude to the hypotenuse AC.




In the question it is given that ABC is right angled and BD is an altitude. Since BD can't be one of the sides of right angled triangle ABC, it has to be the altitude from the right angled vertex to the hypotenuse. Hence B is right angled and BD is the altitude from B to the hypotenuse AC.

Hope its clear!

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A, B, C, and D are distinct points on a plane. If triangle A 10 Sep 2015, 14:20
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For statement 2:

Why can't the product BD*DC = 24 instead of being AB*BC?

Thanks
Cheers
J

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).


(2) says that the product of the non-hypotenuse sides is equal to 24, the non-hypotenuse sides are AB and BC.

Hope it's clear.
Show more


Hi Bunuel,

I have just one question. The reason why i picked "D" was the recollection of a property (which might be wrong looking at the correct ans)
" Perpendicular from the right angle to the hypotenuse divides the triangle in to 2 triangles, resulting in all the THREE triangles being similar to each other. "

In addition, the altitude from the right angle to the hypotenuse bisects the hypotenuse such that hypotenuse length is divided in to two equal lengths, each equal to the length of the altitude.

This is surely not something i have concocted, its just that i do not have the source to quote it.

Please let me know where am i wrong here so that i don't carry forward any wrong, if any, details in the future.

Thanks.
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A, B, C, and D are distinct points on a plane. If triangle A 15 Dec 2015, 10:56
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Quote:


Bunuel, but D doesn't necessarily have to be inside of the triangle right? So what if point D was outside of the triangle vertically above C in your diagram, and equivalent to the original line specified BD, except outside of the triangle? If we kept the same triangle from your diagram, and instead changed the problem to say CD is a height of the triangle with D being vertically about C, then B would still be the right angle and the statements would be inefficient. Am I making a mistake in thinking this?
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We are told that "points A, B, C, and D form a right triangle ABC", so D must be on one of the sides.[/quote]


Hi Bunuel,

We are told that A, B, C and D are on a plane. So, D can lie outside the triangle and hence, statement B will not be sufficient.
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A, B, C, and D are distinct points on a plane. If triangle A 18 Jan 2016, 16:02
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Why can't be the triangle as in the file attached in which case , second statement will not be sufficient.
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A, B, C, and D are distinct points on a plane. If triangle A 18 Jan 2016, 16:05
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Why can't be the triangle as in the file attached in which case , second statement will not be sufficient.
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File comment: Diagram of the triangle
Right angle trangle.gif
Right angle trangle.gif [ 2.61 KiB | Viewed 3203 times ]

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A, B, C, and D are distinct points on a plane. If triangle A 21 Jan 2016, 11:09
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Hello Bunuel,

According to the above mentioned figure, if you consider the triangle ABD then AB is the hypotenuse!! Then how come option B is sufficient!! Kindly explain!!
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A, B, C, and D are distinct points on a plane. If triangle A 21 Jan 2016, 11:12
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Why can't be the triangle as in the file attached in which case , second statement will not be sufficient.
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No, BD must be perpendicular to one of the sides as it is mentioned that "BD is a height of this triangle". The way you have drawn it, you can not consider it to the height of the triangle.
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A, B, C, and D are distinct points on a plane. If triangle A 21 Jan 2016, 11:13
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rocky1988
Hello Bunuel,

According to the above mentioned figure, if you consider the triangle ABD then AB is the hypotenuse!! Then how come option B is sufficient!! Kindly explain!!
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Can you attach or mention what figure are you referring to?
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A, B, C, and D are distinct points on a plane. If triangle A 21 Jan 2016, 23:16
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sudheerarya
Why can't be the triangle as in the file attached in which case , second statement will not be sufficient.

No, BD must be perpendicular to one of the sides as it is mentioned that "BD is a height of this triangle". The way you have drawn it, you can not consider it to the height of the triangle.
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Since the question says BD as height of this triangle, it is a measurement in the plane and the line BD can be parallel to the height of the triangle and need not be the side of the triangle. Why cannot be it taken as a representative measurement in the same plane? When it is parallel to the height of this triangle, it also represents the height of the triangle? I thought every view must be considered while attempting a DS question and selected the answer based on this view. Am I thinking too inappropriately. I feel there shall not be any inambiguity in a GMAT question?
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A, B, C, and D are distinct points on a plane. If triangle A 19 Jun 2017, 04:09
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Bunuel
A, B, C, and D are distinct points on a plane. If triangle ABC is right angled and BD is a height of this triangle, what is the value of AB times BC ?

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 → directly gives us the value of AB*BC. Sufficient.

Answer: B.

Hope it's clear.
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Bunuel:
While I agree with everything where, if in actual exam I get this question, I may be sparing a moment, thinking about another possibility here. When we draw perpendicular BD in the triangle ABC, we technically have 2 hypotenuses now. In this case we really don't have sufficient information to deduce the answer. :shock: I wanted to know what's your take on this.
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A, B, C, and D are distinct points on a plane. If triangle A 19 Jun 2017, 13:20
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23a2012
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enigma123
If distinct points A, B, C, and D form a right triangle ABC with a height BD, what is the value of AB times BC ?

(1) AB = 6.
(2) The product of the non-hypotenuse sides is equal to 24.

I am struggling to understand the solution given in GMATCLub Test. Can you please explain how statement 2 is sufficient to answer this question?


Attachment:
TRIANGLE 1.jpg
If the triangle is as shown in the attachment above then in statement 2 when it says the product of the non hypotenuse sides is 24
then we could also take in ∆ ABD hypotenuse is AB and non hypotenuse sides as AD *BD

also in ∆ BDC hypotenuse is BC and non hypotenuse sides are BD*DC

So statement 2 seems a bit ambiguous.

There are actually 3 right triangles in points ABCD , ∆ ABC, ∆ ABD, ∆ BDC so when they say product of non hypotenuse side
is 24 , we could choose any right triangle among these 3 and take 2 of its non hypotenuse sides .

Why are we assuming that the question is talking of ∆ ABC when it says product of non hypotenuse sides
is 24.

If question had said product of non hypotenuse sides of largest triangle or product of non hypotenuse sides of triangle ABC then 2 could surely be enough.

Please correct me if I am wrong.

Dear Bunuel,I agree with glx and I asked myself the same question above. Why we

assuming that the question is talking about ∆ ABC while the statement 2 did not say that the product of non

hypotenuse sides of largest triangle is 24 ? .
Show more

You're right to bring this up. The language in the problem is vague. On the actual GMAT, if there was any ambiguity, they'd specify what triangle they were referring to.
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A, B, C, and D are distinct points on a plane. If triangle A 14 Mar 2019, 11:44
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We are told that "points A, B, C, and D form a right triangle ABC", so D must be on one of the sides.[/quote]

Actually, as per the question wording - A, B, C, D are on the plane. It does not mention D also forming any part of the right triangle.

Would be great if you could clarify and suggest why D cannot be outside the triangle?
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A, B, C, and D are distinct points on a plane. If triangle A 25 Feb 2022, 19:36
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I agree with previous comments. Why are we assuming point D has to be part of the triangle as this is not specified in the question? It just says that A, B, C, D are on the same plane. I think this is a very important distinction. Thanks
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A, B, C, and D are distinct points on a plane. If triangle A 14 May 2022, 09:26
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earnit
Bunuel
jlgdr
For statement 2:

Why can't the product BD*DC = 24 instead of being AB*BC?

Thanks
Cheers
J

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).


(2) says that the product of the non-hypotenuse sides is equal to 24, the non-hypotenuse sides are AB and BC.

Hope it's clear.


Hi Bunuel,

I have just one question. The reason why i picked "D" was the recollection of a property (which might be wrong looking at the correct ans)
" Perpendicular from the right angle to the hypotenuse divides the triangle in to 2 triangles, resulting in all the THREE triangles being similar to each other. "

In addition, the altitude from the right angle to the hypotenuse bisects the hypotenuse such that hypotenuse length is divided in to two equal lengths, each equal to the length of the altitude.

This is surely not something i have concocted, its just that i do not have the source to quote it.

Please let me know where am i wrong here so that i don't carry forward any wrong, if any, details in the future.

Thanks.
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A, B, C, and D are distinct points on a plane. If triangle A 01 Oct 2023, 00:24
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