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 = 95, c = 125, A = 49°

Accepted Answer

Hey, everyone here we are asked to identify whether the following measurements of two sides and an angle can produce one triangle, two triangles or no triangle. We need to find the missing side lengths and angles of the triangle and express the lengths and angles to one decimal place. Here we have four answer choice options, answer choices A and B state, we only have one triangle with varying angles and side lengths. Then we have answer choice C which states we have two triangles. And lastly, we have answer choice D which states we have no triangle. So here we are given side length A which is units. We then have side length C which is units. And lastly, we have angle A which is 43 degrees. So to begin this problem, we first want to find how many possible triangles we may have. So we first need to recall the law of signs which states that A divided by sign of capital A is equivalent to B divided by sine of capital B, which is equivalent to C divided by sine of capital C. So now we just want to substitute in all of our given information into the formula. And so we have 86 divided by sine of 43 is equal to B divided by sine of capital B, which is equal to 116 divided by sine of capital C. We see that we have no information for lower case or upper case B. So we just want to simplify our equation so that we just have 86 divided by sine of 43 is equal to 116 divided by sine of angle C. And so now we see here that we only have one unknown for our one equation. So we can directly solve for this unknown. So we want to solve for angle C by first multiplying both sides of the equation by sign of capital C. So doing this, we have sine of C multiplied by 86 divided by sine of is equal to 116. And now we just need to divide both sides of the equation by the fraction where we have 86 divided by sine of 43. And so now rewriting, we have sine of C is equivalent to 116 multiplied by sine of 43 all divided by 86. And now to get C alone, we need to take the sign inverse of both sides. And so once again, rewriting, we now have angle C is equal to sine inverse of 116 multiplied by sine of 43 all divided by 86. And now we just need to substitute in or plug in this expression into our calculators to find our angle C and we find that C comes out to 66.9 degrees. And to find its supplementary angle, we just take 180 minus 66. which gives us our other possible angle, which is 113.1 degrees. So we see that C can either be 66.9 degrees or 113.1 degrees. So now we need to use both possibilities and we can start with our first possibility where angle C is 66. degrees. And so we now need to find the rest of our angles and side lengths. So we first need to recall the angle sum property of a triangle which states that angle A plus angle B plus angle C is equal to degrees. And now we just need to substitute in our information. So we have 43 degrees plus unknown angle B plus 66.9 degrees is equal to 180 degrees. Now, just solving for angle B, we just need to subtract 66.9 degrees from both sides as well as subtract 43 degrees from both sides. And we find that angle B is 70.1 degrees and now that we have the angle for B, we can utilize our information from the law of science to find side length B. So we have 86 divided by sine of 43 is equal to B divided by sine of 70.1. And now we just need to solve this expression for B by first multiplying both sides by sine of 70.1. And now doing this to both sides, we get sine of 70. multiplied by 86 divided by sine of 43 is equal to B. And now plugging in this expression into our calculators, we find that side length B is 118.6 units. So for our first triangle, we have that angle C is 66.9 degrees. Angle B is 70.1 degrees and side length B is 118.6 units. Now we're calling from earlier, we know that for its angle C, we might have an additional angle which would be 113.1 degrees. Well, this tells us we may have a second triangle. So once again, we need to utilize the angle sum property of the triangle. So we have angle A which is 43 degrees plus angle B which is unknown plus angle C which is 113.1 degrees and this is equal to 180 degrees. So once again, we need to solve for angle B where in this case, angle B becomes 23.9 degrees. So now that we have angle B, we need to use the law of signs once again to find side length B. So we have 86 divided by sine of 43 is equal to B divided by sine of 23.9 degrees. And just like before, we need to solve this equation for slide length B and we find that B is 51. units. So here we have found that for our second triangle, that angle C is 113.1 degrees, then we have angle B is 23.9 degrees. And lastly, side length B is 51.1 units. So now we have found that we have two possible triangles and we found all of our missing information. So we are left with answer choice C where again, we have two triangles where for our first triangle, we have angle B as 70.1 degrees. Angle C is 66.9 degrees. And side length B is 118.6 units. Then for triangle two, angle B is 23.9 degrees. Angle C is 113.1 degrees. And side length B is 51.1 units. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Key Concepts

Law of Sines

Ambiguous Case of SSA

Triangle Sum Theorem

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