Please show steps and answer, thanks!
Preliminary Information
We know for a fact the lengths of the two parallel sides, the 12in. and 30in. sides. On the left and right sides of the trapezoid you can picture there being a triangle, one on each side. The base for each of the two triangles cannot be assumed to be the same because the angles given are not the same.
12 in.
--------------------------
/ | | \
h1 / | d d | \ h2
/ | | \
---------------------------------
|<x>| 30in. |<y>|
This is roughly the picture I drew so help visualize. We know that the bottom can be rewritten as 30 = 12 + x + y. We can then easily solve this equation for both x and y.
x = 18 - y and y = 18 - x
Since d is the same for both triangles, we solve for each 'd' and then set them equal to each other. We should get:
tan75 = d/(18 - y) for the side with 'x' and tan40 = d/18 - x for the side with y.
Solving these equations for d we get:
d = (18 - y)tan75 and d = (18 - x)tan40
We want d to be in terms of one variable, either x or y. We know that
18 - x = y, so we can sub that in to the second equation for d. We then get:
d = ytan40
Set these equations equal to each other and use algebra to solve for y. You should come to find that y is about 14.6958 and x is about 3.304.
Finding the Perimeter
Now that we know the lengths of the two triangles in the trapezoid, finding the perimeter is much simpler. We still need to find h1 and h2. We know that cos(theta) = adj/hyp. So,
cos75 = x / h1 and cos40 = y/h2
Solving these for h1 and h2 respectively should yield the following:
h1 = 12.7656
h2 = 19.1851
We can now add up all of the sides to get the perimeter.
P = 12 + 30 + h1 + h2
P = 42 + 12.7656 + 19.1851
P = 73.951in.
Finding Area
Finding the area is simpler now that we know some more information about this trapezoid. We need to calculate the value for d so we can use the formula for area of a triangle.
tan75 = d/x
d = xtan75
d = 12.331
You can calculate d using either the side with y or x. You can verify my answer by checking it with the values we found for the side with y.
Now that we know d, we can break up the trapezoid into three shapes. The two triangles on both sides of the trapezoid and the rectangle in the middle. Refer back to the drawing of the sectioned trapezoid at the top of this post.
We can find the area of the triangles using A = (1/2)(bh) and A = hl for the area of the rectangle.
Triangles:
A1 = 0.5(x)(d)
A2 = 0.5(y)(d)
Rectangle:
A3 = 12(d)
You should be able to plug in the values we have found to verify that the area of the trapezoid will be the sum of these smaller areas.
Trapezoid:
A = A1 + A2 + A3
A = 258.9 in.
I hope this all makes sense and you're able to follow my work.