In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

Video transcript

Hey, everyone. Welcome back in this problem. We're asked to find the missing side lengths and angles of the triangle. We're given, we're told to express the length and angles to one decimal place. So we have triangle def, that's given the length of side E is five, the length of side F is 22. And we don't know the length of side D yet. So that's something we'll have to find. We know that angle D is 52 degrees and the other two angles are unknown. We're given four answer choices in this problem options, A through D each containing a different value for those missing side links and the missing angles. And we're gonna come back to these answer choices as we work through the problem. So let's start. And we're actually gonna start by finding the missing side LD and we have two sides and the enclosed angle. So we can use the law of cosines. So to calculate side length D, the law of cosines tells us that D squared, it's going to be equal to E squared was F squared minus two, multiplied by E multiplied by F multiplied by cosine of angle D. So in this case, we're looking for side length D, we know side length E and F and we know angle D. So we have everything we need. Let's go ahead and substitute it. We get D squared. It's going to be equal to five squared plus 22 squared minus two, multiplied by five, multiplied by 22 multiplied by cosine of 52 degrees. Simplify, we have ad squared is equal to 25 plus 484 minus 220 cosine of 52 degrees. And we can work all of this out on our calculator and then take the square root, we're gonna do it in one step. So we avoid some round off error and we get that, that side length D is going to be equal to about 19.327 5574. So we found the missing sideline and we're going round to one decimal place. So we're gonna have 19.3. And if we take a look at our answer choices, option A and option D have that the side length or sorry. Option A and option C have that. The side length is 25.4. This is not what we found. So we can eliminate option A and option C, we're gonna be left with option B or option D which both have this side length, D as 19.3. Yeah, we have the side, all three side links now and we have the corresponding angle to side D as well. So we can actually use the law of signs to find one of these other missing angles. So let's start with angle E and the law of sign tells us that the ratio of the sign of an angle to its side length is equal for each angle in our right. OK. So sine of E divided by E is going to be equal to sine of D divided by D. Again, we know angle D and side D we know side E. So the only unknown in our equation is angle E, we can substitute in what we know. And so let's rearrange first. So sign of angle E is going to be equal to side length of E multiplied by s of angle D divided by side length substituting in our values. We're gonna have that E is equal to the inverse sign five multiplied by sign of 52 degrees divided by that side length. We just found 19.327 5574. And when we work all of this out, we get an angle E how about 11.76 degrees, right. So we have our missing side length, we have one of our missing angles. Let's go back to our triangle and take a look at what we have left. We have one more missing angle left and that is angle F OK. And the last angle is actually the simplest. OK. Recall that the angles in a triangle add up to 180 degrees. OK? Since they add up to 180 degrees and we know the other two angles, we can do a simple calculation to find F. So we know that angle D plus angle E plus angle F must equal 180 degrees. Angle D was given to us as 52 degrees angle E we just found is 11.76 degrees. So 52 degrees plus 11.76 degrees. F must be equal to 180 degrees. We can subtract 52 and 11.76 from both sides. And we're gonna get that F is equal to 116.24 degrees. All right. So now we have all three values we were looking for, we had already narrowed it down to option B or option D. Let's take a look at option B. Option B has an angle E is 65 degrees and angle F is 63 degrees. This is not what we found. So it doesn't look like B is the correct answer. We go to option D and again, D had that side length of 19.3 that we found it also has angle E of 12 degrees and angle F of 116 degrees. And so that is going to be the correct answer here. Option D Thanks everyone for watching. I hope this video helped see you in the next one.