Question

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.

Answer 1

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!

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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 \(\sqrt{17}\)

Answer 6

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

Answer 7

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

Answer 8

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

Answer 9

Seriously!?!

Answer 10

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

Answer 11

Yes, thank you so much!

Answer 12

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

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

oh, yea!! I am wrong!!

Answer 17

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

Answer 18

No worries!!

Answer 19

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

Answer 20

or F towards E the same way

Answer 21

Circle?

Answer 22

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

Answer 23

Lol thank you both for your help and patience!

Answer 24

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

Answer 25

i am guessing

Answer 26

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

Answer 27

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

Answer 28

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

Answer 29

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

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

Answer 30

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

Answer 31

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

Answer 32

that what i said.. :)

Answer 33

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 :)

Answer 34

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

Answer 35

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

Answer 36

17 = (x-4)^2 + (y-6)^2
17 = (x-3)^2 + (y-2)^2

i solved those with a computer,

Answer 37

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

Answer 38

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

Answer 39

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.

Answer 40

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,"....?

Answer 41

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

Answer 42

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!!

Answer 43

draw a picture first

Answer 44

label everything

Answer 45

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

Answer 46

then use the work above

Answer 47

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

Answer 48

*other not multiple

Answer 49

only things i know are fan and testimonial

Answer 50

Testimonial?

Answer 51

@Loser66 did just as much...

Answer 52

hover over name, and buttons are there

Answer 53

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

Answer 54

hah, thanks, ill fan you to

Answer 55

No need!! Good night!!