Answer:
Okay, angle A = 40°. Side a = 9.64. Side b = 11.49.
Explanation:
One step at a time, but this is a longer answer.
1) Out of every triangle existence, its angles always, without exception, add up to 180 degrees. (dramatic, but serious)
2) Knowing that every right angle = 90°, you can write an equation to solve for angle A, your unknown angle.
- 180 - (90+50) = angle A
- 180 - (140) = 40°
3) From here, there's a few ways to solve for the sides, but in my opinion, the fastest and simplest method is to use Law of Sines. If that sounds super complicated and you've never heard of it before, it'll hopefully makes sense at the end of this.
- Some old dead mathematician discovered that, in a triangle, there is an equal, proportionate relationship between the ratio of the sine of one angle divided by its opposing side, and another sine of an angle divided by its opposing side. And this is true for any two angles and sides within that triangle.
- Simply put: sin A/side a = sin B/side B = sin C/side c.
- This is helpful, because as long as you know (or can find out, in our case), two angles and one opposing side to that angle, you're set. (And the law of sines can be used to find an unknown side or an unknown angle.)
4) So, knowing all of this, you can set up a proportion to solve for, let's say, side a. (I attached a picture of this part, in case it seems confusing.)
- In this problem, the right angle is angle C, and equals 90°. Its opposing side is side c, which equals 15.
- So: sin C/side c = sin A/side a.
- sin(90)/15 = sin(40)/side a.
- At this point, we want to isolate side a (which just means to get it by itself on one side of the equation).
- To do this, we want to multiply "side a" on both sides, leaving us with: (side a) x (sin(90))/15 = sin(40)
- Next, we multiply both sides by 15, leaving us with (side a) x (sin(90)) = 15 x (sin(40)).
- After that, we divide both sides by (sin(90)), leaving us with: side a = 15 x (sin(40))/sin(90)).
- Now side a is by itself, and you can type in your calculator: (15 x sin 40)/(sin90). This equals 9.64, which is the length of side a.
5) From here, you can do one of two things to find side B.
- You can do law of sines again, if you feel comfortable with it, and use: sin C/side c = sin B/side b.
- Or, you can use the Pythagorean Theorem, which is: a²+b²=c², and rewrite it so that c²-a²=b². (After you plug in 15 for c and square it, and plug in 9.64 for a and square it, you take the square root of that answer, which gives you the length of side b.) (Another picture attached, if this part is confusing.)
Please let me know if this explanation actually made sense, and I hope it helps!