Step 
In the right triangle ADB
<u>Find the length of the segment AB</u>
Applying the Pythagorean Theorem

we have

substitute the values



Step 
In the right triangle ADB
<u>Find the cosine of the angle BAD</u>
we know that

Step 
In the right triangle ABC
<u>Find the length of the segment AC</u>
we know that




solve for AC

Step 
<u>Find the length of the segment DC</u>
we know that

we have


substitute the values


Step 
<u>Find the length of the segment BC</u>
In the right triangle BDC
Applying the Pythagorean Theorem

we have

substitute the values



therefore
<u>the answer is</u>

Step-by-step explanation:
State you answer in the simplest form
Answer: 5.83 units
Step-by-step explanation:
Sketching the data above,
Length of segment AD can be computed using Pythagoras rule ;
Where segment AD = hypotenuse
Segment AB = 3 = opposite
Sin Θ = opposite / hypotenuse
Sin 31° = 3 / hypotenuse
Hypotenuse × 0.5150380 = 3
Hypotenuse = 3 / 0.5150380
Hypotenuse = 5.8248120 units
Hypotenuse = 5.83 units