To find the length of the diagonal between the points at which the unequal sides meet (part A), we can use the Law of Cosines. Let's refer to the two shorter sides as a and b, and the unknown diagonal as c.
First, let's plug in the given values into the Law of Cosines formula:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where C is the angle formed by the two shorter sides. In this case, C = 115 degrees.
Next, substitute the known values:
c^2 = 20^2 + 20^2 - 2 * 20 * 20 * cos(115)
Simplify and calculate the cosine of 115 degrees:
c^2 = 400 + 400 - 800 * cos(115)
Now, use a calculator to find the cosine of 115 degrees and evaluate the equation:
c^2 ≈ 400 + 400 - 800 * (-0.42261826174)
c^2 ≈ 400 + 400 + 338.094609392
c^2 ≈ 1138.094609392
Finally, take the square root of both sides to find the length of the diagonal:
c ≈ √1138.094609392
c ≈ 33.8 inches (rounded to the nearest tenth of an inch)
Hence, the length of the diagonal between the points at which the unequal sides meet is approximately 33.8 inches.
To find the measure of an angle at which two longer sides meet (part B), we can use the Law of Cosines again. Let's refer to the two longer sides as p and q, and the angle between them as D.
Using the same formula as before:
p^2 = q^2 + c^2 - 2qc * cos(D)
Given that p and q are both 35.0 inches (as mentioned earlier in the question), and we just found that c is approximately 33.8 inches, we can substitute these values into the formula:
35^2 = 35^2 + 33.8^2 - 2 * 35 * 33.8 * cos(D)
Simplifying the equation, we get:
35^2 ≈ 35^2 + 33.8^2 - 2 * 35 * 33.8 * cos(D)
1225 ≈ 1225 + 1142.44 - 2 * 35 * 33.8 * cos(D)
Rearranging the equation, we have:
0 ≈ 1142.44 - 2 * 35 * 33.8 * cos(D)
Substituting the known values into the equation:
0 ≈ 1142.44 - 2 * 35 * 33.8 * cos(D)
Now, we can solve for the angle D by isolating the cos(D) term:
2 * 35 * 33.8 * cos(D) ≈ 1142.44
Multiply both sides by 0.5 and divide by (35 * 33.8):
cos(D) ≈ 1142.44 / (2 * 35 * 33.8)
cos(D) ≈ 1142.44 / 1190
cos(D) ≈ 0.9608
Using the inverse cosine function on a calculator, we can find the measure of angle D:
D ≈ cos^(-1)(0.9608)
D ≈ 15.7772 degrees (rounded to four decimal places)
Therefore, the measure of the angle at which two longer sides meet is approximately 15.8 degrees.