the AC is not a straight line by the way. Just afraid my drawings are not clear
see the modified diagram below. Note that AB = E + F, where E and F are the bases of the two right-triangles we've constructed. we can find their lengths using trig ratios:
EDIT: i forgot about finding the angle CAD. Note that you can find the length of BC in the same way we found AB. once that is found, we can use Pythagoras' theorem to find the length AC. Thus we will know the lengths of all the sides of triangle ACD, and we can use the law of cosines to find any angle in that triangle we want.
Ooopss....
I am very slow in typing. Let me post this anyway.
Let AB be horizontal. So, BC is vertical.
From D, drop a vertical line seggment to AB. Call the intersection as point E.
From D also, draw a horizontal line segment to touch BC. Call the intersection as point F.
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1) To find lenght of BC:
BC = BF +FC
Since BF = DE, then, BC = DE +FC ----**
In right triangle AED,
sin(25deg) = DE / 9
so, DE = 9sin(25deg) = 3.80356 units.
In right triangle CFD,
cos(40deg) = FC / 8
So, FC = 8cos(40deg) = 6.12836 units
therefore, BC = 3.80356 +6.12836 = 9.93192 units long. ------answer.
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To find angle CAD:
In the figure drawn above,
>>>extend ED upwards some more, up to point G ...above hypotenuse AC.
>>>extend FD to the left some more, up to point H ... beyond AC.
angle FCV = angle GDC = 40 degrees
angle EAD = angle HDA = 25 degrees
angle GDH = 90 degrees
so, angle CAD = 40 +90 +25 = 155 degrees. ----answer.