Question

Find the missing part. Use an improper fraction for your answer.

Answer 1

`x=(15)/(2)`
`z=(35113)/(2703)`
`a=(15)/(4)`
`y=(35113)/(5406)`
`b=(45)/(4)`

Explanation:

To solve this, we will have to use multiple methods, including the pythagorean theorem and the 30 60 90 rule.

In the 30 60 90 rule, it is stated that the angle opposing the side with 30 degrees is represented by
`s`, the angle opposing the side with 60 is
`s√(3)` and the angle opposing 90 degree angle is represented by
`2s`. Looking at the whole triangle, we can see the 30, 60 and 90 degrees, and the angle with 90 has an opposing side of 15, which means we have to use 15=2s to find the other sides. When dividing 15 by 2, you get 7.5, which is s. Now we know that the angle opposing 30 is 15/2, which is x. Now that we have "s", we can also solve for z, represented by the 60 degrees. Multiply 7.5 and
`√(3)` to get z=35113/2703.

Now we can solve for the smaller triangles. Since we have already found x, we can use the 30 60 90 method in this too. The x (15/2) is now represented by 2s. Therefore, solve for your new s by doing this: 15/2 = 2s. You should get your 30 degree angle answer (a) with this, which is 15/4. Now that we have to use pythagorean theorem, which is
`a^(2) + b^(2) = c^(2)` to solve for the remaining side (y) of the small triangle. You should get 35113/5406.

Finally we have the last side left: b. Since we have so many sides now, all we have to use is the pythagorean theorem again. Use
`b^2+y^2=z^2` and solve for b. You should get 45/4.