In triangle ΔDEF, m∠D = 45°, m∠E = 63°, and EF = 24 in. What is DE to the nearest tenth of an inch?
30.2 in.
30.2 in.
15.1 in
15.1 in
32.3 in.
32.3 in.
10.5 in.
To find DE, 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 opposite angle is constant.
First, we can find the measure of angle F by subtracting the measures of angles D and E from 180°:
m∠F = 180° - 45° - 63° = 72°
Using the Law of Sines, we can set up the following proportion:
DE / sin(45°) = 24 / sin(72°)
We know that sin(45°) = sqrt(2) / 2 and sin(72°) is approximately 0.9511, so we can substitute these values into the proportion:
DE / (sqrt(2) / 2) = 24 / 0.9511
Simplifying this equation, we get:
DE = (24 * sqrt(2) / 2) / 0.9511
DE ≈ 15.1084
Rounding to the nearest tenth, DE ≈ 15.1 in
Therefore, the answer is 15.1 in.