A, B, C, and D are distinct points on a plane. If triangle A

Question

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 ?

(1) AB=6

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

Bunuel · Accepted Answer

https://gmatclub.com/forum/download/file.php?id=17467 Do you mean that since BD=6, then DC must be 8 and BC must be 10? That's not true. This is a common trap. Knowing that one of the legs is to 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} ... I collected the questions which use this trap: 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 Hope this helps.

Bunuel · Answer

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.

karthiksms · Answer

Could someone please draw the figure for this? thanks

Bunuel · Answer

Here you go: ABC.png Hope it helps.

alphabeta1234 · Answer

Key to this question is "Distinct point" and "Right Triangle". Height BD cannot be AB otherwise the points A=D. So the height BD has to come out of point B (The 90* angle) and intersect AC.

dandarth1 · Answer

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?

Bunuel · Answer

We are told that "points A, B, C, and D form a right triangle ABC", so D must be on one of the sides.

sanjoo · Answer

my question is.. we are asked ab times bc?

bt in statement 2. we have given ab*bc=24? how can we get separate value of both from this?

Bunuel · Answer

We are asked to find AB*BC. (2) says that AB*BC=24. So, we have the value which we wanted to find. Why do we need the values of AB and BC?

sanjoo · Answer

Oh doesnt ab times bc mean ab/bc? or bc/ab? ****! that means i didnt get the question clearly..!!

if ab times bc=ab*bc..fine..then i got the solution !

jlgdr · Answer

For statement 2:

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

Thanks
Cheers
J

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

(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.

ankushbassi · Answer

If the perpendicular is given as 6,that means its a right triangle with base=8 and hypotenuse =10.
Is my understanding wrong?

ankushbassi · Answer

Awesome Bunuel,Thanks a lot.

PathFinder007 · Answer

Hi Bunnel,

I have a small doubt on thic diagram. We know D is 90 degree. But I want to know how B is 90 degree. Is it because we are getting height fron one of the vertices.

Thanks.

ankushbassi · Answer

Since, it is clear that D is the point that has the height.The height of the right triangle always comes form the 90 degree angle,thus B is 90 degree.

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).

qlx · Answer

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.

23a2012 · Answer

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 ? .

A, B, C, and D are distinct points on a plane. If triangle A

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A, B, C, and D are distinct points on a plane. If triangle A

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