Question 890000: In triangle ABC, angle A is 23 degrees, c=29cm, and a=12.2cm. How do I draw two different triangles with these properties and find the measure of angle C in each case?
I've attempted to draw two different triangles but I don't understand, won't I end up with the same angle C in both triangles if they have these same properties?
Thank you for any help you can give!
Hanna
Found 2 solutions by Theo, KMST:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
see this link that discusses the ambiguous case, which is what you have.
http://www.regentsprep.org/Regents/math/algtrig/ATT12/lawofsinesAmbiguous.htm
here's another link from the same website that also discusses the same thing.
http://www.regentsprep.org/Regents/math/geometry/GP4/Ltriangles.htm
here's a picture of your two triangles.
see below the picture for further comments.
angle A is given as equal to 23 degrees and can't change.
side a is opposite angle A and is given as equal to 12.2 and can't change.
side c is opposite angle C and is given as equal to 29 and can't change.
side b is not given and can change.
angle B and angle C are not given and can change.
you can make two triangldes as shown in the diagram.
this is because angle C has two possibilities.
the first possibility is an acute angle.
the second possibility is an obtuse angle.
angle B is determined by the size of angle C.
in both cases, the sum of the angles is equal to 180, so the two triangles are possible.
use the law of sines to find angle C.
a/sin(A) = c/sin(C) becomes 12.2/sin(23) = 29/sin(C)
solve for sin(C) to get:
sin(C) = 29*sin(23)/12.2 = .92878...
angle C is equal to arcsin(.92878...) = 68.2465... degrees which i will round off to 68 degrees for display purposes.
this makes angle B equal to 180 - 23 - 68.2465... = 88.75347... which i will round off to 89 degrees for display purposes.
use the law of sines to find side b.
a/sin(A) = b/sin(B) becomes 12.2 / sin(23) = b/sin(88.75347...
solve for b to get:
b = 12.2 * sin(88.75347...) / sin(23) = 31.2161... which i'll round off to 31.2 for display purposes.
angle C is an acute angle and can possibly be an obtuse angle as well.
the equivalent angle in quadrant 2 is equal to 180 - C which is equal to 180 - 68.2465... which becomes equal to 111.7534... which i'll round off to 112 for display purposes. That becomes angle C'
since the sum of the angles of a triangle is always 180 degrees, that makes angle B' = 180 - A - C' = 45.2465... degrees which i'll round off to 45 degrees for display purposes.
use the law of sines again to find the length of b'
a/sin(A) = b'/sin(B') becomes:
12.2/sin(23) = b'/sin(45.2465...)
solve for b' to get:
b' = 12.2 * sin(45.2465...) / sin(23) = 22.1731... which i'll round off to 22.2 for display purposes.
you have two triangle, each of which has side c = 29, side a = 12.2, and angle A = 23 degrees.
what has changed are side b at 31.2 to side b' at 22.2, angle C at 68 degrees to angle C' at 112 degrees, and angle B at 89 degrees to angle B' at 45 degrees, as shown on the display.
both triangle are possible, given that angle A is 23 degrees and side c is 29 units and side a is 12.2 units.
if two triangles were not possible, more then likely the law of sines would have given you a value for sine that is not possible, such as something greater than 1.
you can also draw an altitude from vertex B to side b and find the measure of that altitude.
if side a is smaller than the altitude than 0 triangles are possible.
if side a is equal to that altitude, than 1 triangle is possible.
if side a is greater than that altitude, than two triangles are possible.
in all the cases above, side a would have to be less than side c.
if side a is greater than side c, then only one triangle is possible.
in your triangle, the alritude came out to be 11.3 or something like that.
side a was greater than that and less than side c so two triangles were possible.
best thing to do if you don't know is to test it out and see if it makes sense.
the following video give you a practial test to see if you can make zero triangles, one triangle, or two triangles.
http://www.youtube.com/watch?v=Io3xZMOrbkQ
Answer by KMST(5328)  (Show Source):
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