Final answer:
To find the length of side e in ΔDEF with given dimensions, we use the Law of Sines and find that e is approximately 19.8 inches to the nearest tenth.
Step-by-step explanation:
To solve for the length of side e in triangle ΔDEF, where we have the length of side f equals 33 inches, the measure of angle m∠F equals 140° and the measure of angle m∠D equals 5°, we can use the Law of Sines. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its corresponding angle is constant. That is:
sin(D) / d = sin(E) / e = sin(F) / f
First, we calculate the measure of angle E using the fact that the sum of the angles in any triangle equals 180 degrees:
m∠E = 180° - m∠D - m∠F = 180° - 5° - 140° = 35°
Then we set up the ratio using the Law of Sines:
sin(D) / d = sin(E) / e
As we don't have the length of side d, we use side f and angle F:
sin(F) / f = sin(E) / e
sin(140°) / 33 = sin(35°) / e
Multiplying both sides by e and then by sin(35°) gives us:
e = f * sin(35°) / sin(140°)
After calculating with a calculator:
e = 33 * sin(35°) / sin(140°) ≈ 19.8 inches
So, the length of side e to the nearest tenth of an inch is approximately 19.8 inches.