list the sides of the triangle in order from longest to shortest. D=86 degrees, E=44 degrees, and F=50 degrees
A. EF, FD, DE
B. FD, DE, EF
C. EF, DE, FD
D. DE, FD, EF
C. EF, DE, FD
To determine the order of the sides of the triangle from longest to shortest, we need to use trigonometry and the Law of Sines.
The Law of Sines states that in a triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
Let's use the letters a, b, and c to represent the lengths of the triangle sides opposite to angles D, E, and F, respectively.
The Law of Sines can be written as:
a/sin(D) = b/sin(E) = c/sin(F)
Given the angles D=86 degrees, E=44 degrees, and F=50 degrees, we can find the order of the sides:
Since D=86 degrees has the largest angle, the side opposite to angle D will be the longest side. Therefore, the longest side is side EF.
So, the order of the sides from longest to shortest is:
EF
Now, let's compare the remaining two sides, FD and DE.
To determine the order between FD and DE, we can compare the ratios FD/sin(E) and DE/sin(D).
Since sin(E) < sin(D) (44 degrees < 86 degrees), we can conclude that FD/sin(E) > DE/sin(D).
By rearranging the equation in the Law of Sines, we can see that the side with a larger ratio is longer. Therefore, FD is longer than DE.
Finally, let's combine all the information:
The order of the sides from longest to shortest is:
EF, FD, DE
Therefore, the correct answer is A. EF, FD, DE.
To determine the sides of the triangle in order from longest to shortest, 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 equal for all three sides. So, let's use the given angles D=86 degrees, E=44 degrees, and F=50 degrees, to find the ratios of the sides.
First, we need to choose a side as a reference. Let's choose side EF as our reference side. So, the ratio for side EF is:
EF / sin(D) = x (let's call it x)
Next, using the Law of Sines and the given angles, we can find the ratios for the other sides:
FD / sin(E) = y (let's call it y)
DE / sin(F) = z (let's call it z)
Now, we need to compare the ratios x, y, and z to determine the lengths of the sides. Remember that the longer the side, the larger the ratio. So, the largest ratio corresponds to the longest side, and the smallest ratio corresponds to the shortest side.
Let's analyze the answer choices:
A. EF, FD, DE
B. FD, DE, EF
C. EF, DE, FD
D. DE, FD, EF
We can start by comparing the ratios for sides EF and FD. If x > y, then EF is longer than FD.
Looking at answer choice A, it states EF, FD, DE. This means that EF is the longest side, which contradicts our comparison. Therefore, we can eliminate answer choice A.
Now, let's compare y and z. If y > z, then FD is longer than DE.
Answer choice B states FD, DE, EF. This satisfies our comparison, as FD is longer than DE. However, we should compare the ratios for sides EF and DE to determine if EF is longer than DE.
Comparing x and z, if x > z, then EF is longer than DE.
Answer choice C states EF, DE, FD. This satisfies our comparison, as EF is longer than DE. Additionally, it satisfies our previous comparison of FD being longer than DE.
Therefore, the correct answer is C. EF, DE, FD, where EF is the longest side, DE is the middle side, and FD is the shortest side.