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In triangle DEF, DF = 16 and m< F = 26. Find FE to the nearest tenth.

14.6

14.38

but it rounds off to 14.4
press: cos (26) x 16

copy and paste it

not enough info

mc070-1.jpg

this is the triangle to the question

To find the length of FE in triangle DEF, we can use the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula for the Law of Cosines is:

c^2 = a^2 + b^2 - 2ab * cos(C)

In our case, we are trying to find the length of FE, so FE will be our side c. The lengths DF and DE will be sides a and b respectively, and the angle F will be angle C.

Let's substitute the given values into the formula:

FE^2 = DF^2 + DE^2 - 2 * DF * DE * cos(F)

FE^2 = 16^2 + DE^2 - 2 * 16 * DE * cos(26)

Now, let's solve for DE:

DE^2 - 2 * 16 * DE * cos(26) = FE^2 - 16^2

DE^2 - 32 * DE * cos(26) + 16^2 - FE^2 = 0

This is a quadratic equation in terms of DE. We can solve it using the quadratic formula:

DE = [ -b ± sqrt(b^2 - 4ac) ] / 2a

In our equation, a = 1, b = -32 * cos(26), and c = 16^2 - FE^2.

Plugging in these values into the quadratic formula, we get:

DE = [ -(-32 * cos(26)) ± sqrt((-32 * cos(26))^2 - 4 * 1 * (16^2 - FE^2)) ] / (2 * 1)

DE = [ 32 * cos(26) ± sqrt(1024 * cos(26)^2 + 4 * (256 - FE^2)) ] / 2

Now, we need to determine which value to choose for DE. Since we are dealing with lengths, DE cannot be negative. Therefore, we can discard the negative sign in front of the square root:

DE = [ 32 * cos(26) + sqrt(1024 * cos(26)^2 + 4 * (256 - FE^2)) ] / 2

Now, we can substitute the value of DE into the equation:

FE^2 = 16^2 + DE^2 - 2 * 16 * DE * cos(26)

FE^2 = 16^2 + ([ 32 * cos(26) + sqrt(1024 * cos(26)^2 + 4 * (256 - FE^2)) ] / 2)^2 - 2 * 16 * ([ 32 * cos(26) + sqrt(1024 * cos(26)^2 + 4 * (256 - FE^2)) ] / 2) * cos(26)

This equation is now in terms of FE. We can solve it to find the length of FE.

To solve this equation, we can use numerical methods or algebraic manipulation. Since it involves a quadratic equation, it might be easier to use numerical methods such as iteration or a graphing calculator to approximate the value of FE to the nearest tenth.

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