Answer:
a = 12.1
A (area) = 66.7
Step-by-step explanation:
Because the right triangle with legs lengths a and 7 has a hypotenuse or 14, you can use the Pythagorean Theorem (I think you know this from the scratch work you have done) and get a^2+49=196, so a^2=147, so a is 7sqrt3, and sqrt3 is approximately 1.732, 7sqrt3 is 7 + 4.9 + 0.21 + 0.014, or 11.9 + 0.224 or 12.124, which is approximately 12.1.
To find the area, we can just do 7sqrt3 times the base length, or 4 + 7 = 11, over 2. This is 77/2 times sqrt3 (1.732). To find this, we can do 38.5 * 1.732, or (40 * 1.732) - (1.5 * 1.732), which is 4 * 17.32 minus (1.5 * 1.732), or 69.28 - 1.732 - 0.866, which is 66.682 or 66.7.
Note: We can use 12.1 as the height, but because it is not fully rounded, you will get a different result (66.55 or 66.6).