SOLUTION: One angle of a triangle is 1/3 the largest angle, but twice the smallest. Find the largest angle.

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Question 205783: One angle of a triangle is 1/3 the largest angle, but twice the smallest. Find the largest angle.
Answer by Targetweek(62) About Me  (Show Source):
You can put this solution on YOUR website!
First we will start out assigning variables
x = shortest side
y = middle side
z = longest side
1.Because all the interior angles of a triangle have to add up to 180 be can write the following equation
- x%2By%2Bz=180
2. Next we write tow additional equations dealing with each variable independantly
3. Because we want to solve for the longest side, z , then we need to write two equations, one where x is by itself (in fancy terms, x in terms of z), and the other where y is by itself (y in terms of z)
- y=%281%2F3%29z
- y=2x
a. Becuase the problem doesn't describe z in terms of x we will have to make one by substituting
b. because y=%281%2F3%29z we can substitue that value into this equation +y=2x
- y=%281%2F3%29z
- 2x+=+%281%2F3%29z
c. Now solve for x
- 2x+=+%281%2F3%29z
- %282x%29%2F2+=+%281%2F3%29z%281%2F2%29
- x+=+%281%2F6%29z
d. Now we have the two equations
- y=%281%2F3%29z
- x+=+%281%2F6%29z
3. Now substitute those equations back into the first equation
- x%2By%2Bz=180
- %28%281%2F6%29z%29+%2B+%28%281%2F3%29z%29+%2B+z+=+180
4. Finally solve for z
- %28%281%2F6%29z%29+%2B+%28%281%2F3%29z%29+%2B+z+=+180
- %281%2F6%29z%2B%282%2F6%29z%2B%286%2F6%29z+=+180
- %289%2F6%29z+=+180
- %286%2F9%29%289%2F6%29z+=+%286%2F9%29180
- z+=+1080%2F9
- z+=+120
5. So the largest angle is 120 degrees
--Check--
y=%281%2F3%29z
y=120%2F3
y=40
x=%281%2F6%29z
x=120%2F6
x=20
20%2B40%2B120+=+180
180+=+180