Answer: x = 8.638156 (approximate)
===================================================
Step-by-step explanation:
Refer to the diagram below.
I've added point labels A,B,C,D to the existing drawing you provided. I also added the label "y" to represent the unknown length AC.
Triangle DAC is isosceles with DC = AC as the two congruent sides. This is shown by the tickmarks.
One useful property of isosceles triangles is that they have congruent base angles. The base angles are opposite the congruent sides.
Since angle ADC = 40, this makes angle DAC = 40 as well.
-----------------------------------------
We'll use this fact to find angle CAB
angle DAB = (angle DAC) + (angle CAB)
80 = (40) + (angle CAB)
angle CAB = 80-40
angle CAB = 40
-----------------------------------------
Move your focus to triangle CAB.
We found or know this already about the triangle
- angle C = 30 (given)
- angle A = 40 (just computed earlier)
Let's find angle B
C+A+B = 180
30+40+B = 180
70+B = 180
B = 180-70
B = 110
Use the law of sines (aka sine rule) to find the value of y, which is side AC.
So,
sin(B)/b = sin(C)/c
sin(B)/y = sin(C)/3
sin(110)/y = sin(30)/3
3*sin(110) = ysin(30)
y = 3*sin(110)/sin(30)
y = 5.638156
which is approximate.
-----------------------------------------
The previous section was all about finding the length of AC. That's approximately 5.638156 units.
We'll move our attention back to triangle DAC.
We know this about the angles
- angle D = 40
- angle A = 40
- angle C = 180-A-D = 180-40-40 = 100
and we determined that side d = 5.638156 which is the length of AC mentioned earlier.
Apply another round of law of sines
sin(D)/d = sin(C)/c
sin(40)/5.638156 = sin(100)/x
xsin(40) = 5.638156*sin(100)
x = 5.638156*sin(100)/sin(40)
x = 8.638156
This value is approximate as well.
Round the values however your teacher instructs.
I used GeoGebra to confirm the x value is correct.