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

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 = 1.4, b = 2.9, A = 142°

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. We want to find the missing side lengths and angles of the triangle and then express the lengths and angles in one decimal place. We are told we have a side A that measures 1.8 A side B that measures 3.7 and an angle A that measures 131 degrees. Our answer choices for answer choice A and B, they each describe one triangle. Answer choice C describes two triangles and answer choice D is if we have no triangle, since we are not told that this is a right triangle, we cannot use the Pythagorean theorem. We cannot assume it's a right triangle. What we will use is the law of signs. So recall that the law of signed states that side A divided by the sign of A would equal side B divided by the sign of B would equal side C divided by the sign of C. In our example, we only have A's and B's. So we're gonna work with side A divided by the sign of A equals side B divided by the sign of B. So side A is 1. divided by the sign of angle A. So the sign of 131 degrees and that'll equal side B. So 3.7 divided by the sign of angle B which is our unknown cross multiplying I have 3.7 multiplied by the sine of degrees equals 1.8 multiplied by the sign of B. We're gonna divide both sides by 1.8 to get the sign of B alone. So I have 3.7 multiplied by the sine of 131 degrees divided by 1.8 will equal the sign of B. I want to place this in a calculator to find out the value. And I'm going to use parentheses around the 3.7 multiplied by the sine of 131 degrees. And then divide by 1.8. And when I do so this will come out to approximately 1.5513 would have to equal the sign of B. Now, before I try to find what B is as an angle, I recall that sine cannot be greater. So it has to be less than, or equal to one. Since 1.55 is greater than one, we have no triangle here because it does not work out to be within the range of sign. So this is answer choice D no triangle. Have a nice day.

Key Concepts

Law of Sines

Triangle Inequality Theorem

Ambiguous Case of SSA

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