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Math Expert V
Joined: 02 Sep 2009
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19 00:00

Difficulty:   35% (medium)

Question Stats: 66% (01:38) correct 34% (02:10) wrong based on 205 sessions

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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 of triangle ABC is equal to 24.

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Math Expert V
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Official Solution:

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 of triangle ABC is equal to 24 $$\rightarrow$$ directly gives us the value of AB*BC. Sufficient.

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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.

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?

Math Expert V
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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.

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?

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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first option says AB =6
doesnt it imply that the sides are 6,8,10?
Math Expert V
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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:
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if-distinct-points-a-b-c-and-d-form-a-right-triangle-abc-129328.html

Hope this helps.
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Bunuel wrote:
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.

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?

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). Bunuel -- in the figure above isn't BD an altitude and AB the height of triangle ABC??
Math Expert V
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efforts wrote:

Bunuel -- in the figure above isn't BD an altitude and AB the height of triangle ABC??

Hi,
there is no difference in " altitude and height"..
" altitude or height" is the shortest distance from a point to the line..
so here BD is the altitude/height, when AC is the base and AB is the alt/height when base is BC...
Hope it clears the query..
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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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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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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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wow I can't believe this question tricked me so hard. Thank you for this question. It really makes me read a LOT more carefully.
Current Student Joined: 21 Apr 2016
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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...
>> !!!

You do not have the required permissions to view the files attached to this post.

Math Expert V
Joined: 02 Sep 2009
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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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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!
Math Expert V
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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:
Attachment:
alt2.gif

>> !!!

You do not have the required permissions to view the files attached to this post.

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Bunuel wrote:
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.

-----

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:
Attachment:
alt2.gif
[/quote]

'VERTICES' is the magic word!! Gotcha, Thanks!  Intern  Joined: 26 Apr 2016
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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.
Math Expert V
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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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banty1987 wrote:
I think this is a high-quality question and I don't agree with the explanation. In statement (2) it say's product of the non-hypotenuse sides. from the figure in the explanation even AB and BC are hypotenuse to triangle's ADB & BDC respectively. Please explain if I am wrong. in the question it does not say hypotenuse as AC only.

Please read the discussion on previous pages.
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