Answer: x = 9.981675
The value is approximate. Round it however you need.
==========================================================
Step-by-step explanation:
I'll add point labels A,B,C,D
Refer to the diagram below to see how the points are laid out. It doesn't matter how you label the points. I'm doing this to describe the triangles we'll be working with.
For now, focus on triangle CAD, which is the smaller triangle at the bottom.
Use the Law of Sines to determine the value of y, which is the angle I marked in red.
sin(A)/a = sin(C)/c
sin(40)/8 = sin(y)/9
sin(y) = 9*sin(40)/8
sin(y) = 0.723136
y = arcsin(0.723136)
y = 46.314005
For triangle CAD, angle C is roughly 46.314005 degrees.
In other words, angle ACB is about 46.314005 degrees.
----------------------------
Keep your focus on triangle CAD.
The interior angles must add to 180
A+C+D = 180
A+y+z = 180
40 + 46.314005 + z = 180
86.314005+z = 180
z = 180 - 86.314005
z = 93.685995
This is the approximate measure of angle ADC in purple.
It's very close to 90 degrees, but not quite there exactly. This is one reason why the diagram alone can be misleading.
----------------------------
We'll keep our attention on triangle CAD. The last thing we need from this small triangle is the side of AC, aka side d since it's opposite angle D.
Use the law of sines again
sin(A)/a = sin(D)/d
sin(40)/8 = sin(93.685995)/d
d*sin(40) = 8sin(93.685995)
d = 8sin(93.685995)/sin(40)
d = 12.420045
We've wrapped up everything to say about triangle CAD. We can move on.
----------------------------
There's a lot to keep track of. Here's a summary so far. I'll only summarize what is needed from here on out.
- Angle ACB = y = 46.314005 degrees (red)
- Side AC = d = 12.420045
Those decimal values are approximate.
The value of z was useful earlier, but we don't need it anymore.
----------------------------
Turn your attention to triangle ABC.
Because ABC is a right triangle (50+40 = 90), this means we can use one of the six trig ratios to determine the hypotenuse x+8 based on a known acute angle and known leg of the triangle.
Knowing the leg AC and angle ACB is sufficient to find the hypotenuse BC.
Use the cosine ratio to tie everything together like so
cos(angle) = adjacent/hypotenuse
cos(C) = AC/BC
cos(46.314005) = 12.420045/BC
BC*cos(46.314005) = 12.420045
BC = 12.420045/cos(46.314005)
BC = 17.981675
This is the approximate length of the hypotenuse of right triangle ABC.
The last step to do really is to set this equal to x+8 and solve for x.
x+8 = 17.981675
x = 17.981675-8
x = 9.981675 is our approximate final answer
Round this however you need to, or however your teacher instructs.
Once again, there's a lot to keep track of. Feel free to ask any further questions if you're stuck anywhere, or if one of my steps doesn't make sense.