Without using the law of sines, explain why no triangle ABC ca...

Question

Without using the law of sines, explain why no triangle ABC can exist that satisfies A = 103° 20', a = 14.6 ft, b = 20.4 ft.

Accepted Answer

Welcome back everyone in this problem. Without using the law of signs, we want to choose the correct reason for the non existence of the triangle with the measurement that says angle A is 117 degrees and 40 minutes. Side A is 27.9 centimeters long and side B is 36.2 centimeters long. For our answer choices A says the triangle is non existent because angle A is an obtuse angle and side B is larger than side A or longer than side A B says angle A is an obtuse angle and side A is longer than side BC says A is an acute angle and side A is longer than side B and D says A is an acute angle and side B is longer than side A. Now, we know that our triangle is non existent that's given here. But let's assume that we were trying to draw our triangle. Then with all the law of science, we're trying to show how we know based on the diagram or based on the drawing or the reasoning that it could not exist. So let's try to draw a sketch of that. So we have our angle A here, which is an obtuse angle. So we know it's going to be greater than 90 degrees in our sketch. OK. So let's say that this is Anger A and then we're going to have our triangle here. OK? Where this is Anger A, then we could say Anger B is here. OK? And we could also say that Angus C is right here. Now, we know that Anger Air is 117 degrees and 40 minutes. OK? We also know that side A is going to be opposite to Anger A and it says here, that side A is 27.9 centimeters. OK? Let me put A equals here. And we also know that side B here is going to be opposite to B which is going to be equal to 36.2 centimeters. So if it exited, it would look something like that. But what is it about this triangle that tells us that it cannot exist? Well, we know that the summer of triangles, the sum of angles in a triangle equals 180 degrees. OK? So we know that by the angle, some property of the triangle, if A plus B plus C equals 180 degrees, then angles B and C, the summer angles B and C are going to be equal to 180 degrees minus 117 degrees and 40 minutes, which would be equal to 62 degrees. OK. 62 degrees and 20 minutes. So the sum of B and C are less than 90 degrees and A is greater than 90 degrees. So we can see here that A is an obtuse angle. Now, since A is the largest angle in the triangle, because the sum of the other two angles are still less than the value of A, then the side opposite to A must be the greatest side as we can see here in our diagram. That makes sense because if it's the greatest angle opposite to it must be the longest side. But in our triangle here, it says A is 27.9 centimeters and B is 36.2 centimeters. So based on this information, it would, it would appear that B is greater than A. So that's A, that, that has to be a contradiction because B cannot be greater than A B has to be less than A as A is the longest side. So we know that this triangle does not exist because A is an obvious angle and is saying that B is longer than A. Therefore, the correct choice would be answer choice. A thanks a lot for watching everyone. I hope this video helped.

Without using the law of sines, explain why no triangle ABC can exist that satisfies A = 103° 20', a = 14.6 ft, b = 20.4 ft.

Key Concepts

Triangle Inequality Theorem

Angle-Side Relationships in Triangles

Sine Rule and Angle Constraints

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