Solve each triangle ABC.

A = 39.70°, C = 30.3...

Question

Solve each triangle ABC.

A = 39.70°, C = 30.35°, b = 39.74 m

Accepted Answer

Hello, today we are going to be finding the missing side links and angles of triangle ABC. We are also going to label this information on a generic triangle. So we are told that angle A is equal to 31.65 degrees. We are also told that angle C is equal to 26.21 degrees. And we are told that S length B is equal to 49.24 m. So how can we go about solving for the remaining angles and S links? Well, we can first start by solving for angle B, we can solve for angle B by using the sum angle property for triangles. This states that the sum of all three angles within a triangle must equal to 180 degrees. So we are going to take angle A which is 31.65 degrees at angle C which is 26.21 degrees and add angle B and this sum should equal to 180 degrees. So let's go ahead and solve for angle B 31.65 plus 26.21 will give us an angle of 57.9 degrees. So we have 57.9 plus B is equal to 180 degrees. Subtracting 57.9 to both sides of the equation will give us angle B is equal to 122.14 degrees. This is going to be the first missing piece of information that we need. And we will also be labeling this on our triangle. So we know that angle B is 122.14 degrees. Now how can we solve for the sings A and C? In order to do this, we are going to use the law of signs. The law sign states that we have three equivalent ratios. We have sign of angle A divided by S length A is equal to sine of angle B divided by sine length B is equal to sine of angle C divided by S length C. What we can do is we can take any two of the ratios and set them equal to each other in order to solve further sins A and C, let's go ahead and start by solving for the side length A. Now we know that we have angle B and the side length B, we have angle A but we don't have the side length A. So what we can do is we can take the ratio with respect to B and set it equal to the ratio with respect to A and use that to solve for the side length of A. So we will have sign of A divided by sine length A is equal to sine of B divided by side length B. In order to solve for A, we will cross multiply the two ratios. This will leave us with a multiplied by sine of B is equal to si length B multiplied by sine of A. And in order to solve for A, we're going to divide both sides of this equation by sine of B. This will leave us with A is equal to SI length B multiplied by sine of A divided by sine of B. We know that the side length B is equal to 49.24. The angle A is equal to 31.65 degrees. And we sold for angle B to be 122.14 degrees. Plugging in these values will give us 49.24 multiplied by sign of 31.65 divided by S of 122.14. At this step, I would recommend using a calculator to help you solve this value plugging in this value into a calculator will give you the value of S length A equal to 30.51 m. This is going to be the second piece of missing information we are looking for. And we can also label this measurement on our generic triangle now that we have the values of A and the values of B we will need to solve for the side length C. Now, just like before we can take the ratio with respect to B and set it equal to the ratio with respect to C. So now we are going to solve for the side length C. In order to do this, we will take sign of B divided by S length of B and set it equal to sine of C divided by the S length C just like before we are going to cross multiply the ratios in order to solve for C that will leave us with C multiplied by S of B is equal to the S length B multiplied by sine of C. We will divide both sides of this equation by sine of B leaving us with C is equal to the si length B multiplied by sine of angle C divided by sine of angle B. So plugging in the values we know that the S length B is 49.24. We know that the Anglesey was given to us as 26.25. So we will have sign of 26.25 divided by S of B which is sign of 122.14. And plugging in this value into a calculator will give us the side length C is equal to 25.68 m. This is going to be the final piece of information that we need for the triangle. So just to recap, we use the sum angle property to solve for angle B which is 122.14 degrees. We use the law of signs to solve for side length A which came out to be 30.51 m. And we use the law of signs to solve for the side length C which is 25.68 m. With that being said, 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.

Solve each triangle ABC.

A = 39.70°, C = 30.35°, b = 39.74 m

Key Concepts

Triangle Sum Theorem

Law of Sines

Side-Angle Relationships

Watch next