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Geometry Questions

by gmatusa2010 » Sat Nov 20, 2010 9:01 pm
A few quick geometry theory questions. Can someone please help.

1) Pathagorean triples:

a) For a given angle, let's say you know one of the length is 3 and the other length is 4, and they are both not the hypotenuse, is that sufficient to conclude that it is part of a pathagorean triple and thus a right angle?

b) What if you know length of hypotenuse is 5 and the other is 4, can u conclude that it is a pathagorean triple and thus a right triangle?


2)30/60/90 and 45/45/90 Triangles:

a) If you know one angle and the length of two sides (correspondingly) that fit the criteria for one of these two triangle, is that sufficient to conclude that: It is a 30/60/90 or 45/45/90?

Ex1: Triangle ABC, angle ABC is 30 degree, its opposite is 1 and the hypotenuse is 2, can you conclude anything?

Ex2: Triangle ABC, angle ABC is 30 degree, its opposite is 1 and another length is squareroot (3), can you conclude anything?

3) Other triangles:

a) If you know one angle, and the length of two sides, is it possible to find the length of the 3rd side? (You don't know whether it is 90 degree). I ran into this problem on a DS question. The OA suggest that you CAN but doesnt explain how since its a DS question as long as you can thats enough. Can someone expound on the theory that allows you to do so?
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by shovan85 » Sat Nov 20, 2010 9:23 pm
gmatusa2010 wrote:A few quick geometry theory questions. Can someone please help.

1) Pathagorean triples:

a) For a given angle, let's say you know one of the length is 3 and the other length is 4, and they are both not the hypotenuse, is that sufficient to conclude that it is part of a pathagorean triple and thus a right angle?
No. When one is 3, other side is 4, the remaining side of the triangle can be any value between 1 and 7 (excluding). Thus we cannot conclude if the same is a right angle triangle.
When you get 3-4-5 then and only then it is a right angle triangle.
gmatusa2010 wrote: b) What if you know length of hypotenuse is 5 and the other is 4, can u conclude that it is a pathagorean triple and thus a right triangle?
You must either know that the angle opposite to hypotenuse is 90 or Remaining side length is 3 to say if it is Right-angle triangle .
gmatusa2010 wrote: 2)30/60/90 and 45/45/90 Triangles:

a) If you know one angle and the length of two sides (correspondingly) that fit the criteria for one of these two triangle, is that sufficient to conclude that: It is a 30/60/90 or 45/45/90?

Ex1: Triangle ABC, angle ABC is 30 degree, its opposite is 1 and the hypotenuse is 2, can you conclude anything?

Ex2: Triangle ABC, angle ABC is 30 degree, its opposite is 1 and another length is squareroot (3), can you conclude anything?
I do not think you can directly conclude so. You must be sure that you are looking in to a 30-60-90 Right angle triangle. Just by knowing one angle 30 and length two sides you cannot say that whether it is a right-angle triangle.

Though you can verify or try the Pythagorean theorem, but you cannot conclude so.

More info: https://www.themathpage.com/atrig/30-60-90-triangle.htm
gmatusa2010 wrote: 3) Other triangles:

a) If you know one angle, and the length of two sides, is it possible to find the length of the 3rd side? (You don't know whether it is 90 degree). I ran into this problem on a DS question. The OA suggest that you CAN but doesnt explain how since its a DS question as long as you can thats enough. Can someone expound on the theory that allows you to do so?
See Expert Rahul's post below.
Last edited by shovan85 on Sat Nov 20, 2010 10:04 pm, edited 1 time in total.
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by gmatusa2010 » Sat Nov 20, 2010 9:42 pm
Picture: https://magoosh.question-images.s3.amazo ... figure.PNG
(Let me know if that doesnt work. Don't know what the policy is to posting beatthegmat practice quesiton. I will remove if someone tells me it violates)

Question:



If CD = 6, what is the length of BC?

(1) BD= 6Squareroot(3)

(2) x = 60


OA: D


The nuance might be in the logic of the question. The answer explanation seems to imply that 1 is sufficient because theres only 1 possible length for the triangle with length 6, 6root(3), and 30 degree. But I thought the question asks what is the length of BC. Eventhough there's only one possible length, you CANNOT calculate whatever that number is from the info. given. (Maybe you can with Trig but thats out of the scope of the test).












shovan85 wrote:
gmatusa2010 wrote:A few quick geometry theory questions. Can someone please help.

1) Pathagorean triples:

a) For a given angle, let's say you know one of the length is 3 and the other length is 4, and they are both not the hypotenuse, is that sufficient to conclude that it is part of a pathagorean triple and thus a right angle?
No. When one is 3, other side is 4, the remaining side of the triangle can be any value between 1 and 7 (excluding). Thus we cannot conclude if the same is a right angle triangle.
When you get 3-4-5 then and only then it is a right angle triangle.
gmatusa2010 wrote: b) What if you know length of hypotenuse is 5 and the other is 4, can u conclude that it is a pathagorean triple and thus a right triangle?
You must either know that the angle opposite to hypotenuse is 90 or Remaining side length is 3 to say if it is Right-angle triangle .
gmatusa2010 wrote: 2)30/60/90 and 45/45/90 Triangles:

a) If you know one angle and the length of two sides (correspondingly) that fit the criteria for one of these two triangle, is that sufficient to conclude that: It is a 30/60/90 or 45/45/90?

Ex1: Triangle ABC, angle ABC is 30 degree, its opposite is 1 and the hypotenuse is 2, can you conclude anything?

Ex2: Triangle ABC, angle ABC is 30 degree, its opposite is 1 and another length is squareroot (3), can you conclude anything?
I do not think you can directly conclude so. You must be sure that you are looking in to a 30-60-90 Right angle triangle. Just by knowing one angle 30 and length two sides you cannot say that whether it is a right-angle triangle.

Though you can verify or try the Pythagorean theorem, but you cannot conclude so.

More info: https://www.themathpage.com/atrig/30-60-90-triangle.htm
gmatusa2010 wrote: 3) Other triangles:

a) If you know one angle, and the length of two sides, is it possible to find the length of the 3rd side? (You don't know whether it is 90 degree). I ran into this problem on a DS question. The OA suggest that you CAN but doesnt explain how since its a DS question as long as you can thats enough. Can someone expound on the theory that allows you to do so?
I believe only in the cases of Right-angle Triangle, Isosceles triangles, and Equilateral triangles you can find the length of 3rd side. Please post the DS question if you have.
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by Rahul@gurome » Sat Nov 20, 2010 9:45 pm
gmatusa2010 wrote:1) Pathagorean triples:

a) For a given angle, let's say you know one of the length is 3 and the other length is 4, and they are both not the hypotenuse, is that sufficient to conclude that it is part of a pathagorean triple and thus a right angle?

b) What if you know length of hypotenuse is 5 and the other is 4, can u conclude that it is a pathagorean triple and thus a right triangle?
The term 'hypotenuse' is only defined for right-angled triangles. Thus, if it is mentioned that the triangle has a hypotenuse, it is enough to conclude that the triangle is right-angled. If in general you're asking the questions without using the term 'hypotenuse' then,
  • 1. a) Yes, they are a part of a Pythagorean triple. But that doesn't mean the triangle is right-angled.
    1. b) Yes, they are a part of a Pythagorean triple. But that doesn't mean the triangle is right-angled.
gmatusa2010 wrote:2)30/60/90 and 45/45/90 Triangles:

a) If you know one angle and the length of two sides (correspondingly) that fit the criteria for one of these two triangle, is that sufficient to conclude that: It is a 30/60/90 or 45/45/90?

Ex1: Triangle ABC, angle ABC is 30 degree, its opposite is 1 and the hypotenuse is 2, can you conclude anything?

Ex2: Triangle ABC, angle ABC is 30 degree, its opposite is 1 and another length is squareroot (3), can you conclude anything?
I have no idea what do you mean by "correspondingly", but if any triangle fit the criteria to be a 45-45-90 or 30-60-90 triangle, then obviously they are! But your examples tell me you wanted to mean something else!
  • Ex1: Once again, if hypotenuse is involved it has to be right-angled triangle. Otherwise, no we can't conclude anything.
    Ex2: No, we can't.
gmatusa2010 wrote:3) Other triangles:

a) If you know one angle, and the length of two sides, is it possible to find the length of the 3rd side? (You don't know whether it is 90 degree). I ran into this problem on a DS question. The OA suggest that you CAN but doesnt explain how since its a DS question as long as you can thats enough. Can someone expound on the theory that allows you to do so?
Yes, if we know only one angle and the length of two sides, then it is possible to find the length of the 3rd side. But not in all case. The two sides must be the sides adjacent to the known angle. Otherwise we can't. If they are adjacent sides to the given angle, the 3rd side can be found by applying trigonometry.
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by gmatusa2010 » Sat Nov 20, 2010 10:00 pm
Rahul, I thought the definition of pathagorean triple is X^2+Y^2= C^2.


"The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a2 + b2 = c2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a = b = 1 and c = √2 is right, but (1, 1, √2) is not a Pythagorean triple because √2 is not an integer. "

https://en.wikipedia.org/wiki/Pythagorean_theorem

Rahul@gurome wrote:
gmatusa2010 wrote:1) Pathagorean triples:

a) For a given angle, let's say you know one of the length is 3 and the other length is 4, and they are both not the hypotenuse, is that sufficient to conclude that it is part of a pathagorean triple and thus a right angle?

b) What if you know length of hypotenuse is 5 and the other is 4, can u conclude that it is a pathagorean triple and thus a right triangle?
The term 'hypotenuse' is only defined for right-angled triangles. Thus, if it is mentioned that the triangle has a hypotenuse, it is enough to conclude that the triangle is right-angled. If in general you're asking the questions without using the term 'hypotenuse' then,
  • 1. a) Yes, they are a part of a Pythagorean triple. But that doesn't mean the triangle is right-angled.
    1. b) Yes, they are a part of a Pythagorean triple. But that doesn't mean the triangle is right-angled.
gmatusa2010 wrote:2)30/60/90 and 45/45/90 Triangles:

a) If you know one angle and the length of two sides (correspondingly) that fit the criteria for one of these two triangle, is that sufficient to conclude that: It is a 30/60/90 or 45/45/90?

Ex1: Triangle ABC, angle ABC is 30 degree, its opposite is 1 and the hypotenuse is 2, can you conclude anything?

Ex2: Triangle ABC, angle ABC is 30 degree, its opposite is 1 and another length is squareroot (3), can you conclude anything?
I have no idea what do you mean by "correspondingly", but if any triangle fit the criteria to be a 45-45-90 or 30-60-90 triangle, then obviously they are! But your examples tell me you wanted to mean something else!
  • Ex1: Once again, if hypotenuse is involved it has to be right-angled triangle. Otherwise, no we can't conclude anything.
    Ex2: No, we can't.
gmatusa2010 wrote:3) Other triangles:

a) If you know one angle, and the length of two sides, is it possible to find the length of the 3rd side? (You don't know whether it is 90 degree). I ran into this problem on a DS question. The OA suggest that you CAN but doesnt explain how since its a DS question as long as you can thats enough. Can someone expound on the theory that allows you to do so?
Yes, if we know only one angle and the length of two sides, then it is possible to find the length of the 3rd side. But not in all case. The two sides must be the sides adjacent to the known angle. Otherwise we can't. If they are adjacent sides to the given angle, the 3rd side can be found by applying trigonometry.
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by Rahul@gurome » Sat Nov 20, 2010 10:08 pm
gmatusa2010 wrote:Rahul, I thought the definition of pathagorean triple is X^2+Y^2= C^2.

"The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a2 + b2 = c2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a = b = 1 and c = √2 is right, but (1, 1, √2) is not a Pythagorean triple because √2 is not an integer. "

https://en.wikipedia.org/wiki/Pythagorean_theorem
Yes, that's perfect!
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by gmatusa2010 » Sat Nov 20, 2010 10:17 pm
Can you explain why it can be a pathagorean triple BUT not a 90 degree angle? According to the definition Pathegorean Theorem only applies to 90 degree angles.
Rahul@gurome wrote:
gmatusa2010 wrote:Rahul, I thought the definition of pathagorean triple is X^2+Y^2= C^2.

"The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a2 + b2 = c2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a = b = 1 and c = √2 is right, but (1, 1, √2) is not a Pythagorean triple because √2 is not an integer. "

https://en.wikipedia.org/wiki/Pythagorean_theorem
Yes, that's perfect!
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by Rahul@gurome » Sat Nov 20, 2010 10:33 pm
gmatusa2010 wrote:Can you explain why it can be a pathagorean triple BUT not a 90 degree angle? According to the definition Pathegorean Theorem only applies to 90 degree angles.
If you're talking about example 1.(a) of your question, i.e. if you know that two sides are 3 and 4, then they are part of a Pythagorean triple because there exists an integer c, for which (3)² + (4)² = c². In fact c = 5. This doesn't mean the length of the other side of the triangle is 5! The length of the other side of the triangle (say x) may be anything that satisfies the inequality, (4 - 3) < x < (4 + 3) => 1 < x < 7

Sides of the right-angled triangle and Pythagorean triples are not the same thing! While sides are a geometrical concept, Pythagorean triples are a concept of number theory. If two sides of a triangle are part of Pythagorean triple, then this doesn't necessarily mean the triangle is right-angled.
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by shovan85 » Sat Nov 20, 2010 10:33 pm
gmatusa2010 wrote: Question:
If CD = 6, what is the length of BC?

(1) BD= 6Squareroot(3)

(2) x = 60
Please ignore my answer to your last question. What Rahul has mentioned is absolutely correct. Let me apply what he has mentioned and try to solve the above problem.

(1) BD= 6Squareroot(3)

Applying trigonometry to triangle CDX, (Refer below attached diagram)

DX = CD cos 30 = 6*(Squareroot(3)/2) = 3 Squareroot(3)
CX = CD sin 30 = 6 * (1/2) = 3

BX = BD - DX = 6Squareroot(3) - 3 Squareroot(3) = 3 Squareroot(3)

Consider triangle BCX,

BC = Squareroot(CX^2 + BX^2) = Squareroot( 3^2 + [3 Squareroot(3)]^2 ) = Squareroot (9+27) = 6.

Thus, Sufficient.

(2) x = 60

if x = 60 then angle BCD = 120
As angle CDB = 30 then angle CBD = 180 - (120+30) = 30
Thus BC = CD = 6. (Isosceles )

Thus, Sufficient.
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by gmatusa2010 » Sun Nov 21, 2010 1:17 am
Wow. Thanks guys. Crystal clear now. Geometry is so hard to prove to yourself unless you have a pretty "advance" knowledge.
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