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Condition for a right angle triangle

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Condition for a right angle triangle [#permalink] New post 03 Mar 2013, 23:13
If a triangle has an angle 30 degree and two other sides (Not the one making the angle) have lengths in the ratio of 1:2, is it a sufficient condition for the triangle to be right angle? (e.g. Triangle ABC has angle B = 30 degree and AC:BC = 1:2, Is it a sufficient condition for angle A to be 90 degree?)

It appears to be true, but I am unable to prove it with the knowledge of geometry I have. I have my GMAT approaching in 10 days. Would appreciate if someone could prove it at the soonest. Thanks.
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Re: Condition for a right angle triangle [#permalink] New post 03 Mar 2013, 23:47
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randomalgo wrote:
If a triangle has an angle 30 degree and two other sides (Not the one making the angle) have lengths in the ratio of 1:2, is it a sufficient condition for the triangle to be right angle? (e.g. Triangle ABC has angle B = 30 degree and AC:BC = 1:2, Is it a sufficient condition for angle A to be 90 degree?)

It appears to be true, but I am unable to prove it with the knowledge of geometry I have. I have my GMAT approaching in 10 days. Would appreciate if someone could prove it at the soonest. Thanks.


Yes it is. The law of sines says :

a/sinA = b/sinB = c/sinC

Thus, here, we have 2x/sinA = x/sin30

or sinA = sin30*2 = 0.5*2 = 1.

Thus, A = 90 degrees.

I have assumed the triangle described by you in the query.
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Re: Condition for a right angle triangle [#permalink] New post 03 Mar 2013, 23:52
Thanks Vinay for the answer, you got the question right. but Isn't Sine Rule applicable only to right angle triangles? In that case, you assumed that one angle is 90 degree and applied the rule only to find out which one.
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Re: Condition for a right angle triangle [#permalink] New post 04 Mar 2013, 00:04
Expert's post
randomalgo wrote:
Thanks Vinay for the answer, you got the question right. but Isn't Sine Rule applicable only to right angle triangles? In that case, you assumed that one angle is 90 degree and applied the rule only to find out which one.



No. It is true for any valid triangle.
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Re: Condition for a right angle triangle   [#permalink] 04 Mar 2013, 00:04
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Condition for a right angle triangle

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