In Exercises 17–32, two sides and an angle (SSA) of a triangle...

Question

In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively.
a = 7, b = 28, A = 12°

a = 7, b = 28, A = 12°

Accepted Answer

Hello, everyone. We are asked to identify whether the following measurements of two sides in an angle can produce one triangle, two triangles or no triangles. If they produce triangles, we'll find the missing side lengths and angles and express the lengths and angles in one decimal place. We are given side A equals 12 side B equals 35. And the angle at A is 18 degrees. In our answer, choices A and B are the measures of one triangle. C says we have two triangles and its associated measures and D would be for no triangles because we do not know that this is a right triangle. We are going to use the law of signs. We're called that the law of signs says the side A divided by the sign of angle A is proportional to the side B divided by the sign of angle B and is also equal to side C divided by the sign of angle C. So knowing this, I'm gonna try to find what is an angle that I don't already know. So we have A and B in our given information. So we're gonna start with side A which is 12 divided by the sine of angle A which is 18 degrees. And that's going to equal the measure of side B 35 divided by the sign of angle B which will be our unknown for now using cross multiplication, I get 35 multiplied by the sign of degrees equals 12 multiplied by the sign of B. I'm going to divide both sides by 12. And I'm gonna see if 35 sine of 18 degrees divided by 12 is greater than one. Because if it's greater than one or less than one, I have no triangles. So let's find out what it is. When I do that out, I get that the sign of B equals 0.9012996. So that's less than one that falls into our range. So we can take the inverse sign of 0.9012996. That way we can find a value for B and when we do so we find that B will be 64.3 degrees. So we now know the angle at B in addition to the angle at a using the angle sum theorem of a triangle. Sorry, the triangle angle sum theorem, we know that all of our angles must add up to 100 80 degrees. So to find angles C I can do degrees minus 18 degrees minus 64.3 degrees. And when I do so the angle at sea is 97. degrees. So we have at least one triangle, we need to find one more thing. And that is the measure of side C. Since this is not a right triangle, I am not gonna be able to use the Pythagorean theorem, I'm going back to the law of signs. So I'm gonna use a divided by sine of a. So 12 divided by the sine of 18 degrees equals are unknown of side C divided by the sign of Anglesey. We just found so 97.7 degrees. And again, when I cross multiply, I get 12 multiplied by the sine of 97.7 degrees equals C multiplied by the sign of 18 degrees. I'll divide both sides by the sine of 18 degrees. And I am definitely using a calculator to find these values. And when I do so I get that lower case C or side C is equal to 38.48. Since we want this to be one decimal place would be 38.5 units. So we know at the very least we have one triangle where angle C is 97.7 degrees, angle B is 64.3 degrees. And the side see is 38.5 units. So this could be answer choice A or C at this point. Since when we found the sign of B to be 0.9012996. That was positive. I can say that the angle B occurs when sin is positive, sin is positive in both the 1st and 2nd quadrants. So I'm gonna use quadrant two now and I'm gonna take the angle I found for the first B so 64. and to find the measure of angle B in the second quadrant, I'm gonna do 180 degrees minus 64.3. And this will tell us that in the second quadrant angle B would have a measure of 115.7 degrees. So I'm gonna put a little sub two on that just so we keep track of which one's which. So now we have to do the process again to find the value of angle C and since this angle does exist and does look like it works, we are going to have two triangles as our answer. So to find the value of angle C, we would again do 180 degrees minus angle A which is degrees minus angle B which is 115.7 degrees. And that comes out to angle C being 46. degrees. So now we must find our missing side, which is side C again, I'm gonna use the law of signs like I did in the previous triangle. So I have a divided by the sign of angle A. So 18 degrees equals C divided by the sign of Anglesey. So 46.3 degrees cross multiplying I get 12 multiplied by the sign of 46.3 degrees equals C multiplied by the sign of 18 degrees, dividing both sides by the sine of 18 degrees. And again, I'm gonna use a calculator to find this exact value. I end up with C equaling 28.7. So that's C sub two. Um But we want that rounded to one decimal place. So that'd be 28.1 units. So we do have two triangles. So that would be answer choice. C the first triangle has angle B at 64. degrees angle C at 97.7 degrees and side C at 38.5 units. Triangle two has angle B at 115.7 degrees angle C at 46.3 degrees and side C at 28.1 units have a nice day.

Key Concepts

Law of Sines

Ambiguous Case (SSA) in Triangle Solutions

Triangle Inequality and Angle-Side Relationships

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