Solve each right triangle. When two sides are given, give angles ...

Question

Solve each right triangle. When two sides are given, give angles in degrees and minutes. See Examples 1 and 2.

Accepted Answer

Hey, everyone in this problem, we're asked to find the missing side length and the missing angles of the right triangle. We're given and we're told to express the angle in degrees and minutes. The triangle we're given, we have a right angle at Q. The distance from Q to R is 10.4 centimeters. The distance from R to P is little Q. The distance from P to Q is 12.6 centimeters. We're given four answer choices A through D each containing different values for the missing angles. P and R in the missing side. A little Q. And we're gonna come back to those answer choices as we work through this problem. Now, let's start with the missing side Q and we're gonna start there because that's gonna be the simplest thing to solve for. So let's start for the missing side Q. And because we have a right triangle and we already know two of the other sides. We can use Pythagorean theorem. We call that Pythagorean theorem tells us that the hypotenuse which in this case is Q squared is going to be equal to the sum of the other side squared. So R squared plus Peace Court. And when we substitute in our values, we get that Q squared is going to be equal to 12.6 centimeters squared plus 10.4 centimeters squared. Simplifying on the right hand side, Q squared is going to be equal to 266 0.92 centimeters squared. And finally, we can take the square root and we're gonna take the positive root because we're looking at a distance, let me get 16.3377 approximately centimeters. OK. So that's the missing side length you and if we look at our answer choices, option A and option C have 16.3 centimeters, which is what we found. Um Option D also has that correct value of 16.3 centimeters. Option B has 13.3 centimeters. So we can eliminate option B that does not have the correct side links. So we're down to three possible answer choices. And now we have these two missing angles to find. So angle R and angle P, let's start with angle P. Now we have all three side links now, but the side links we're gonna choose to use are gonna be the ones that were given in the problem. This just helps us avoid some round off error. OK. So if we use the ones given in the problem, we don't have to deal with the round off error we generated when we found Q. So we have the adjacent and the opposite side of our angle that we're interested in. So we're gonna relate them through tangent, they call that tangent. However, angle P is going to be the opposite side divided by the adjacent side, which in this case is going to be 10.4 centimeters divided by 12.6 centimeters. We can take the inverse tangent on both sides. And what we get is that our angle P must be equal to 39.54 degrees. OK. So we found our angle in degrees. But remember that the answer wants it in degrees and minutes. So we're gonna have to take the fraction of a degree we found and convert that into minutes. So we have 0.54 of a degree and recall that in every one degree, we have 60 minutes. So we're gonna multiply by 60 minutes divided by one degree, the unit of degrees divides out and we're left with 32.4 minutes. Now the answer choices around to the nearest minute. And so when we put this all together, we get that P is going to be 39 degrees and 32 minutes. They were rounding off that 0.4. OK. So we have angle P now 39 degrees and 32 minutes. And now we're gonna switch over and find angle Q oh sorry angle R. Now, for angle R, we could do it the same way we've done here, we can use tangent, we can relate the sides or we can use the fact that we have a triangle, we know the angles add up to 180 degrees. OK. So let's go ahead and do that. So we know that R is going to be 180 degrees minus the 90 degree angle. We have minus angle P that we just found 39 degrees and 32 minutes. And when we do this subtraction, we find that R is going to be 50 degrees and 28 minutes. So using this idea that the sum of the angles in a triangle add up to 180 degrees makes that third angle much easier to find. There's a lot less calculation to do. So now we have everything we were looking for. We found that angle P is 39 degrees and 32 minutes. Angle R is 50 degrees and 28 minutes and side Q is about 16.3 centimeters. When we compare this to the answer choices we have, we can see that this matches with option D. Thanks everyone for watching. I hope this video helped see you in the next one.

Solve each right triangle. When two sides are given, give angles in degrees and minutes. See Examples 1 and 2.

Key Concepts

Right Triangle Properties

Trigonometric Ratios

Angle Measurement

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