Determine the number of triangles ABC possible with the given ...

Question

Determine the number of triangles ABC possible with the given parts.

a = 31, b = 26, B = 48°

Accepted Answer

Welcome back. Everyone. In this problem, we want to find a number of possible triangles with the measurements. That side A equals 52 units. Side B is equals 46 and angle B equals 59 degrees. For our answer choices A says there are one, there's one triangle B says two and C says there are none. Now, if we want to find a number of possible triangles, first, let's try to draw at least one of the triangles to get an idea of what it may look like. So let's start with one that says we have an angle B being 59 degrees. OK? So let's say that angle B is here, that's 59 degrees. And for this triangle, OK? It must have anger A and it also would have anger C where the side A is opposite to the angle A, so the side A is 52. OK? So let's say A, let's put A equals 52 here and the side B is opposite to angle B. So the side B equals 46. Now, if we are going to figure out if or how many of those triangles exist, we'd have to think about the different scenarios for the angles, the possible angles because we know that the sum of angles in a triangle equal 180 degrees. So if we can find the other angles in the triangle, and then check if there are any other possible values, then we should be able to figure out if any other triangles exist. So let's start by trying to find the Angus four or sketch that's given here. OK. Now, if we think about it for this triangle, we have a pair of sides that are opposite and the values are known. And if we want to find another side, let's say that we were trying to find the value of a, then notice here that we don't know anger. A but we know the side that's opposite to it. Now, what can help us in a situation like this where we have a known pair of side and anger? Well, we can use the law of science because recall that the love of science tells us that the ratio of sides to the sine of the opposite angle in a triangle is the same throughout the triangle. So by the law of sines, then it's telling us that the ratio of the sine of A to its opposite side A is equal to the sine of B to its opposite side B which is equal to the sine of C to its opposite side C. So here in our problem we, we have, we know anger B, we know the value of side B. We also know the value of side A but we don't know the value of anger A. So we can solve for anger A using the law of sign. So solving for A, I'm sorry for anger A OK, then the sign of A divided by 52 the value of A is going to be equal to the sine of B which is 59 degrees divided by the value of B which is 46. Therefore, the sign of A is going to be equal to 52 sine 59 divided by 46 we divided, we multiplied both sides by 52. And thus, to figure out A, we could find the S inverse. In other words, the question is the sign of what angle equals our expression. Well, to figure it out, we can find the inverse sign of our expression. So A equals the inverse sine of 52 S 59 degrees divided by 46. And when we work that out, we get the value of A to be equal to, let's round it to two decimal places. So rather the value of A is approximately equal, approximately equal to 75.69 degrees. OK. So that's our value for A, know that we have a value for A and B. Let's try to find the value for side C. Now, if we know two angles. So now we know A is uh 75.69. OK. And let me put that in blue, you know what? Just to follow that's 75.69 degrees. Then we can solve for angle sea because this tells us then if we have two angles and we want to fight the third angle in a triangle. Recall that the sum of angles in a triangle must sum to 180 degrees. So that tells us then that solving co C loving for angle C C must be equal to 180 degrees minus 59 degrees plus 75.69 degrees. Not when we work that out, we get c to be equal to 45.31 degrees. So this is one possible, we, we know so far that there's one possible triangle. Now, how can we know if there's another triangle? Well, we can do that by subtracting our value for a, from 180 degrees. And if we summit with angle B and it's less than 180 degrees, that means it's possible for another triangle to exist. If it's greater than 180 degrees, then the sum of angles in the triangle would have to be also greater than 180 degrees. But that's impossible. OK. So let's first try to do that. No, the other possible value for air, then the other value for a is going to be 180 degrees minus 75.69 degrees, which would be equal to 104.31 degrees. Therefore, if we add the angles, we know so far A and B A plus B is going to be equal to 104.31 degrees plus 59 degrees, which would be equal to 163.31 degrees. And this answer is less than 180 degrees. So since the sum of the two angles is less than 180 degrees A could also have been 104.31 degrees. Therefore, there are two possible triangles for our sides here. For our measurements. It could either be 75.69 degrees or it could have been 104.31 degrees. The correct answer is b thanks a lot for watching everyone. I hope this video helped.

Determine the number of triangles ABC possible with the given parts.

a = 31, b = 26, B = 48°

Key Concepts

Law of Sines

Ambiguous Case (SSA Condition)

Triangle Inequality and Angle Constraints

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