Which one of the following sets of data does not determine a u...

Question

Which one of the following sets of data does not determine a unique triangle?

a. A = 50°, b = 21, a = 19

b. A = 45°, b = 10, a = 12

c. A = 130°, b = 4, a = 7

d. A = 30°, b = 8, a = 4

Accepted Answer

Hello, today we are going to be choosing the option that will not represent a unique triangle. Now notice that each of the options given to us have an angle A, a side length B and a side length A. In order to figure out which option does not represent a unique triangle, we are going to be using the law of signs. Now recall that the law of sine states that we have three equivalent ratios. We have sine of angle A divided by SI length A equal to sine of angle B divided by side length B equal to sine of angle C divided by SI length C. What we are going to do is we are going to take the ratio with respect to A and set it equal to the ratios with respect to B. And the reason why we are doing this is because we want to solve for the angle B. Once we get the angle B, we will be able to solve for the angles within the triangle for each of the options and use the angles to determine if each of the options will not represent a unique triangle. Let's go ahead and start by checking option A. So option A tells us that we have an angle A equal to 48 degrees. We have a side length B equal to 29 and we have a side length A equal to 24. Using the law of signs, we will have sine of angle A divided by side length, A equal to sine of angle B divided by S length B. We want to go ahead and solve for the angle B to do this. Let's go ahead and cross multiply the ratios that will leave us with side length A multiplied by sine of angle B equal to side length B multiplied by sine of angle A. We will divide both sides of this equation by the side length A and that will leave us with sine of B equal to the S length B multiplied by sine of A divided by the side length A. In order to solve for the angle B, we are going to be taking the sine inverse of both sides of this equation that will leave us with the angle B is equal to the sine inverse of S length B multiplied by sine of angle A divided by the side length A. Now go ahead and take a note of this equation we just came up with for the angle B. The reason for this is because we are going to reuse this equation to check for the angle B for options BC and D continuing with option A, let's go ahead and plug in the given side links and angle that will leave us with the sine inverse of 29 multiplied by sine of 48 degrees divided by 24. Now, I would recommend plugging in a uh plugging in this value into a calculator to get the value for angle B. When plugging into a calculator, the value for angle B that we get is going to be 64 degrees. Now, because we got this from a sign inverse, we want to take the alternate version of the angle B, the alternate version will equal to 100 and 80 degrees minus 64 degrees. This will give us an angle of 116 degrees. So this is going to be the measurement for angle B. Now, what we want to do is we want to take the sum of the angle B with the sum of the angle A and as long as the sum is less than 180 degrees, this is going to give us a valid measurement for a triangle. So we are going to add angle A to 116 degrees, which is 48 degrees, 116 plus 48 will give us a final sum of 164 degrees. And since this angle is less than 180 degrees, this means that we, this will represent two triangles, not just one unique triangle. So, so far option A is a valid option for a solution. Now, let's go ahead and check the remaining options. Let's go ahead and check B. Now, now B tells us that the measurement of angle A is 38 degrees. The side length B is 18 and the side length A is 23. We are going to use the same equation that we came up with earlier to sulfur the angle B. So we will have the angle B is equal to the sine inverse of 18 multiplied by sine of 38 degrees divided by 23. After plugging this into a calculator, we will get an approximate value of 29 degrees. And we want to take the alternate version of angle B. So we will be subtracting 29 degrees from 180. This will give us a measurement of 151 degrees. Now, in this case, our angle A was 38 degrees. If we add angle A to the angle B, we will have 151 degrees plus 38 degrees, which will give us a total of 189 degrees. This value is greater than 180 degrees. So this is not a valid option for a triangle. So option B will not be our solution. We will go ahead and check C. Now, now C tells us that the angle A is equal to 111 degrees. Side length B is nine and side length A is 13. Using the equation from earlier, we will have the angle B is equal to the sine inverse of nine multiplied by s of 111 divided by 13 using a calculator. This will give us an approximate value of 40 degrees. And we will take the alternate option for angle B by subtracting 40 degrees from 100 and 80 degrees. 180 minus 40 will give us the value of 140 degrees. And if we add angle A which is 100 and 11 degrees to this volume, we will end up with 250 degrees. This value is greater than 180 degrees. So this option is not valid for a triangle. Finally, we will check option D, option D tells us that the angle A is 54 degrees. The side length B is 19 and the side length A is 29. Plugging this into the equation for the angle B will give us the angle B is equal to the sine inverse of 19 multiplied by sine of 54 divided by 29. After plugging this into a calculator, we would get an approximate angle of 32 degrees and taking the alternate version of the angle will be 180 minus 32 which will give us the final angle of 148 degrees, we're going to add angle A to this value. So we have 148 plus 30 plus 54 apologies, which will give us a final angle of 202 degrees. This is greater than 180 degrees. So this is not a valid option for a triangle. So since option A tells us that this is going to give us two triangles and one, not one unique triangle, that means that the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Which one of the following sets of data does not determine a unique triangle?
a. A = 50°, b = 21, a = 19
b. A = 45°, b = 10, a = 12
c. A = 130°, b = 4, a = 7
d. A = 30°, b = 8, a = 4

Key Concepts

Triangle Congruence Criteria

Ambiguous Case of the Law of Sines

Sum of Angles in a Triangle

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