SOLUTION:

Question 1203444:
Found 5 solutions by josgarithmetic, MathLover1, ikleyn, Theo, greenestamps:
Answer by josgarithmetic(39799)   (Show Source): You can put this solution on YOUR website!
Recognize and understand:

You should have no trouble.
Answer by MathLover1(20855)   (Show Source): You can put this solution on YOUR website!

if is the midpoint of , then
If and , then

what is

Answer by ikleyn(53763)   (Show Source): You can put this solution on YOUR website!
.
E is the midpoint of DF. If EF = x + 9 and DF = 3x + 9, what is DF
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        The solution by @MathLover1 is incorrect.
        I came to bring a correct solution.

Then 2(x+9) = 3x + 9, which implies, step by step 2x + 18 = 3x + 9 18 - 9 = 3x - 2x 9 = x DF = 3x + 9 = 3*9 + 9 = 27 + 9 = 36. ANSWER. DF = 36.

Solved (correctly).

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In his post, @Theo, without any visible reasons, came with the other interpretation
and solved different problem from what was in your original post.

Solving different problems is not prohibited;
but you, the visitor, should understand that different problem is irrelevant to your original problem.

Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
you mentioned DF twice.
one of those menioned should be DE, rather than DF.
this leads to two interpretations.
i'll do both and you can decide which one you meant.

first interpretation:

E is the midpoint of DF. If EF = x + 9 and DE = 3x + 9, what is DF?

since E is the midpoint, then DE must be equal to EF.
this means that 3x + 9 = x + 9
subtract x from both sides of the equation and subtract 9 from both sides of the equation to get 2x = 0
solce for x to get x = 0
when x = 0, x + 9 = 9 and 3x + 9 = 9
this means that DE = 9 and EF = 9
since DF is the sum of its two line segments, then DF must be 18.

second interpretation.

E is the midpoint of DF. If EF = x + 9 and DF = 3x + 9, what is DE?

since E is the midpoint, then DE must be equal to EF.
this means that DE = EF.
since EF = x + 9, then DE must also be equal to x + 9.
you get:
DE + EF = DF becomes x + 9 + x + 9 = 3x + 9
combine like terms to get:
2x + 18 = 3x + 9
subtract 2x from both sides of the equation and subtract x from both sides of the equation to get:
9 = x
when x = 9, EF = 18
snce DE = EF, then DE = 18 also
you get:
18 + 18 = 3x + 9
combine like terms to get 36 = 3x + 9
since x = 9, you get 36 = 27 + 9 which becomes 36 = 36.
this confirms x = 9.

in the first interpretation, you were asked what is DF and the solution was DF = 18 and x = 0.

in the second interpretation, you werre asked what is DE and the solution was DE = 18 and x = 9.

we'll go through both interpretations again and see if the solutions make sense.

first interpretation:

E is the midpoint of DF. If EF = x + 9 and DE = 3x + 9, what is DF?

solution was DF = 18 and x = 0 *****

when x = 0, EF = 9 and DE = 9, you get 9 + 9 = 18, which is correct.

second interpretation:

E is the midpoint of DF. If EF = x + 9 and DF = 3x + 9, what is DE?

solution was DE = 18 and x = 9 *****

when x = 9 and DE = 18, you get 18 + 18 = 36, which is correct.

take your pick as to which of the interpretations you meant to provide.
your solution is the one associated with that interpretation.

Answer by greenestamps(13334)   (Show Source): You can put this solution on YOUR website!

Apparently one of the tutors in preparing his response erased the original question....

But tutor @ikleyn repeated the question in her response, so I can see what the question was.

If E is the midpoint of DF, then DE (= DF - EF) is equal to EF. Given EF = x+9 and DF = 3x+9, we have

(3x+9)-(x+9) = x+9
2x = x+9
x = 9

And the length of DF is 3x+9 = 3(9)+9 = 27+9 = 36.

ANSWER: DF = 36