Question

How many triangles are there that satisfy the conditions a = 14, b = 2, a = 66°?

Possible answers

impossible to determine


2


0


1

Answer 1

How many triangles can you draw that have those elements?

Answer 2

1?

Answer 3

Or do I need the other side to draw it?

Answer 4

No, you just have to connect the two endpoints...the only issue is whether or not it is possible to do so at all. In this case, it is.

Answer 5

Oh okai thank you c:

Answer 6

looks like 13.3125 is the length of the other side...

Answer 7

I just needed to determine how many triangles it could make given those points.

Answer 8

Also, you might want to use this as a guide:

|14 - 2| < c < |14 + 2|
12 < c < 16

You know the third side has to be between 12 and 16

Answer 9

Thank you.

Answer 10

^That result pretty much let's you know there's only one possible side. and you confirm the value using law of sines.

Answer 11

Now suppose you had two given sides:

a = 16 and b = 13

|16 - 13| < c < |16 + 13|

3 < c < 29

Now the range is wide enough for you to consider that there might be two possible lengths: one short, and one long.

Answer 12

for the third side, c.

Answer 13

And you confirm with law of sines to make sure the angles exist.

Answer 14

How can you confirm that with the law of sines?

Answer 15

Because if the angles do not exist, you get a false value back

Answer 16

when you try to solve using law of sines

Answer 17

Oh okai I see, I forgot I had the angle for that.

Answer 18

For example, suppose you have

a = 3.5, b = 5
A = 51 degrees

You consider the third side might be

|5 - 3.5| < c < |3.5 + 5|
1.5 < c < 8.5

But then you check with law of sines, and get back a false result...Then you know that such a triangle is not possible.

Answer 19

Oh so I have to check the math to determine if the triangle is possible not just look at the picture?

Answer 20

That's right.

Now suppose you had

a = 3
b = 9
A = 115 degrees

Now A is obtuse, but side a is only 3 while b > a

You can just stop right there because no way is this possible. The side a cannot be the shortest side yet have an obtuse angle as its corresponding angle.

Answer 21

Oh I get it! So for this specific problem, there would be 2 answers possible?

Answer 22

Since it has to be between 12 and 16?

Answer 23

No, 12 and 16 are too close to each other.

If you had something like what I was showing you earlier where 3 < c < 29

Then you can consider that there may be two possible triangles. Notice that 3 is short while 29 is long.

Answer 24

You might want to re-read what I wrote above.

Answer 25

Oh okai so if the answers I get are only for a long side or only for a short side, than that means there is only 1 answer available?

Answer 26

If I that is right than I get it c:

Answer 27

You use it as a guide mainly.

Answer 28

Ah I see.

Answer 29

If you have something like 12 < c < 16, then it's pretty likely that there's only one possible or none depending on what law of sines tell you about the existence of the angle.

If you have something like 3 < c < 6, then again, you consider the same reasoning.

If you have something like 3 < c < 29...this is a huge range...so there could be two possible triangles. Double check with law of sines.

Answer 30

Okai I understand I believe c:

Answer 31

@you-me-at-six
I am wondering - Have you studied the Law of Cosines?

Answer 32

Honestly? No, I am just learning as I do my exams because I won't need to take another math class for another year, I just don't have time to read my book that has not helped me learn anything, so basically I'm just taking a question and simply trying to figure out what I can do with it. So no I have no studied the Law of Cosines nor have I studied anything else in pre-calc or trig.

Answer 33

The Law of Cosines is an algebraic way to determine the number of triangles possible in a scenario such as the one you posted. I had in mind to work the posted problem using the Law of Cosines up to the point where it is evident how many triangles are possible. That's why I asked.

Answer 34

Oh I thought you were insulting my intelligence cx and okai I see why you asked.

Answer 35

I am respectful of intelligence and would never seek to insult anyone.