Determine if triangle DEF with coordinates D (3, 2), E (4, 6), and F (7, 3) is an equilateral triangle. Use evidence to support your claim. If it is not an equilateral triangle, what changes can be made to make it equilateral? Be specific.

- anonymous

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- anonymous

I used the distance formula for each segment in triangle DEF and have come to the conclusion that the triangle is not an equilateral triangle. The distance for Segments DE and DF are sq.rt.17, but the distance for Segment EF is sq.rt. 18. I am having difficulty answering the second part of the question, "If it is not an equilateral triangle, what changes can be made to make it equilateral? Be specific." Please provide guidance as to how to go about answering this question. Thank you!

- anonymous

Find the distance between D and E to determine the length of Segment DE.
D = (3,2) and E = (4,6)
Step 1: d = sq.rt. (x2 - x1)^2 + (y2 - y1)^2
Step 2: d = sq.rt. (4 - 3)^2 + (6 - 2)^2
Step 3: d = sq.rt. (1)^2 + (4)^2
Step 4: d = sq.rt. 1 + 16
Step 5: d = sq.rt. 17
Find the distance between E and F to determine the length of Segment EF.
E = (4,6) and F = (7,3)
Step 1: d = sq.rt. (x2 - x1)^2 + (y2 - y1)^2
Step 2: d = sq.rt. (7 - 4)^2 + (3 - 6)^2
Step 3: d = sq.rt. (3)^2 + (-3)^2
Step 4: d = sq.rt. 9 + 9
Step 5: d = sq.rt. 18
Find the distance between D and F to determine the length of Segment DF.
Step 1: d = sq.rt. (x2 - x1)^2 + (y2 - y1)^2
Step 2: d = sq.rt. (7 - 3)^2 + (3 - 2)^2
Step 3: d = sq.rt. (4)^2 + (1)^2
Step 4: d = sq.rt 16 + 1
Step 5: d = sq.rt. 17

- DanJS

i guess, sqrt(18) - sqrt(17) needs to be taken off EF, and recalculate the position of one of the points

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## More answers

- anonymous

I agree, but I have tried different combinations and have failed to find an alternative point.

- Loser66

By distance formula , you have \(EF =\sqrt{(7-4)^2+(3-6)^2}=\sqrt {18}\)
Only one way you can do is to make it down to \(\sqrt{17}\)

- anonymous

That is what I thought before, but I got points off because I didn't give an alternative point.

- Loser66

by changing the location of E.
F cannot be changed because it relates to D

- Loser66

if E( 6, 7) then you get the required triangle.

- anonymous

Seriously!?!

- Loser66

Read the problem, they say : by what changes??

- anonymous

Yes, thank you so much!

- Loser66

my change is the point E from (4, 6) to (6,7)

- DanJS

E or F can move in a circle around point D, right

- Loser66

@DanJS It is a good idea but I don't think they use the orbit concept here.

- anonymous

I plugged in 6,7 yet still got different measures for each distance. Did I do something wrong?

- Loser66

oh, yea!! I am wrong!!

- Loser66

since if you change E, then DE changes also.
OMG. I am terribly sorry.

- anonymous

No worries!!

- DanJS

I dont see how that has to happen, if you move E , DE can stay the same length, the radius of the circle, when you move E on an Arc towards F

- DanJS

or F towards E the same way

- anonymous

Circle?

- Loser66

I know how to do!! Thanks God. He gives me a chance to fix my mistake. hehehe.

- anonymous

Lol thank you both for your help and patience!

- DanJS

Circle centered at D, radius root(17), you just swing F towards E or E towards F, till the chord EF is root17

- DanJS

i am guessing

- DanJS

or set up two distance formula , and use (x,y) for F, and figure out x and y, possibly

- Loser66

OK.
\(DF=\sqrt{17}\)

- anonymous

I tried your second option, but every time I changed a coordinate, I failed to get the same measure for each segment.

- Loser66

\(DE=\sqrt{(x_E-3)^2+(y_E-2)^2)}\)
\(EF=\sqrt{(7-x_E)^2+(3-y_E)^2)}\)

- Loser66

and we want DE = EF, so, solve for it!!

- Loser66

\(DE = EF = \sqrt{17}\)

- DanJS

that what i said.. :)

- Loser66

ok, take over, please. hehehe.. I tried to fix my mistake but if you know how to do, please, finish the stuff. I am not good at explaining. Please :)

- anonymous

So the change or different coordinate is (x,y)?

- DanJS

there are two possible points for F, one on each side of the Y axis, i think

- DanJS

17 = (x-4)^2 + (y-6)^2
17 = (x-3)^2 + (y-2)^2
i solved those with a computer,

- anonymous

So D still equals (3,2)...E still equals (4,6). What does F = ?

- DanJS

\[F(x,y) = (\frac{ 4\sqrt{3} + 7 }{ 2 } , \frac{ 8-\sqrt{3} }{ 2 }) \approx (6.9641, 3.1340)\]

- anonymous

Woah...that is very precise. I think I may have gone about answering this question the wrong way because I doubt my teacher would have expected such an answer.

- anonymous

Do you think I should answer using your original suggestion, "i guess, sqrt(18) - sqrt(17) needs to be taken off EF, and recalculate the position of one of the points,"....?

- DanJS

\[FD = FE = \sqrt{17}\] that what those 2 distance equations above are

- anonymous

Thank you for your help and time! If you could, would you please help me formulate my final answer. If not, I totally understand because I have already taken up so much of your time!!

- DanJS

draw a picture first

- DanJS

label everything

- DanJS

then maybe, move F up a little, and relable that F' point (x,y), EF' = root (17) to make an equilateral triangle

- DanJS

then use the work above

- anonymous

Okay, that is perfect! Thanks again. Can I give you multiple awards...other than just a medal?

- anonymous

*other not multiple

- DanJS

only things i know are fan and testimonial

- anonymous

Testimonial?

- DanJS

@Loser66 did just as much...

- DanJS

hover over name, and buttons are there

- anonymous

Okay, I gave you a medal, became a fan and submitted a testimonial. Thanks!!

- DanJS

hah, thanks, ill fan you to

- anonymous

No need!! Good night!!

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