Question 252312
step 1: draw a parallelogram slanting right and label the base 12 and the slant height 8. 
step 2: draw a diagonal from top left to bottom right.
step 3: construct an altitude from the top left vertex down to the base. call that "h".
step 4: call the left side segment "a" and the right side segment "b". a + b = 12.
step 5: we have 2 pythagorean theorems: (i) {{{a^2 + h^2 = 8^2}}}; (ii) {{{b^2 + h^2 = 10^2}}}.
step 6: by subtracting the two, we get (iii) {{{b^2 - a^2 = 36}}}.
step 7: find values for a and b such that (b+a)(b-a) = 36. If b= 9 and a = 3, we have the values.
By using either (i) or (ii), this also gives us the height, h,  as sqrt(55).
step 8: draw the other diagonal. Since we are dealing with a parallelogram, opposite sides have the same slope. So, the base grows from 12 to 15.
step 9: we apply another pythagorean theorem to our new triangle. (iv) {{{15^2 + sqrt(55)^2 = d^2}}}
step 10: solve for d. we get d^2 = 225 + 55 = 280. So, d ~ 16.7332.

Hope that helps.