Question

Hey I was wondering if you can help me out with the question?

Answer 1

We have a figure made of four right triangles.

Now, we need to find the missing sides for each triangle to get the side x.

Let's use the triangle ABE:

AB represents for this triangle the adjacent side and we need to find the

opposite side BE.

To find the value for BE, we need to use trigonometric functions. The functions must involve the side that we know and the side that we want to find. Therefore:

Tan = opposite side/ adjacent side

Replacing the values:

Tan 45 = BE/AB

Tan 45 = BE/ 10in

Solve for BE:

BE = Tan 45* 10in

BE = 10in

Now, for triangle BEF:

BE represents the adjacent side and BF the hypotenuse.

Therefore, we use the next trigonometric function

cos = adjacent side/ hypotenuse

Replacing the values:

Cos 60 = BE/ BF

Cos 60 = 10/ BF

Solve for BF

BF = 10/Cos 60

Then

BF = 20in

Now, for triangle BFC:

The angle B is equal to 180. We have a right angle and an angle of 60 degrees. Therefore, for m∠B

m∠B = 90+60 + a = 180

Solve for a:

150 + a = 180-150

a = 30

Where a is the inside angle b for the triangle BEF.

In this triangle, BF represents the hypotenuse and we need to find the opposite side CF.

Hence, we use the next trigonometric function:

Sin = opposite side/ hypotenuse

Replacing the values:

Sin 30 = CF/BF

Sin 30 = CF/20

Solve for CF:

CF = 10 in

Finally, for triangle CFD:

CF represents the adjacent side and we need to find the hypotenuse DF = x.

Hence, we use the next trigonometric function:

Cos = adjacent side/ hypotenuse

Replacing the values:

Cos 30 = CF/x

Cos 30 = 10/x

Solve for x:

x = 10/Cos 30

Then:

`x=\frac{20\sqrt[]{3}}{3}`

Therefore, the correct answer is the third option.