Show that (4,5,2) ,(1,7,3),and (2,4,5) are vertices of an equilaterial triangle.

Please teach me how to solve this question. Thank you very much.

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- Dec 21st 2006, 09:20 PMJenny20A question, distance formula, triangle
Show that (4,5,2) ,(1,7,3),and (2,4,5) are vertices of an equilaterial triangle.

Please teach me how to solve this question. Thank you very much. - Dec 21st 2006, 11:23 PMCaptainBlack
Compute the (square)distances between the points, if they are equal, then you have

an equilateral triangle.

first two:

D^2=(4-1)^2+(5-7)^2+(2-3)^2=9+4+1=14

first and third:

D^2=(4-2)^2+(5-4)^2+(2-5)^2=4+1+9=14

second and third:

D^2=(1-2)^2+(7-4)^2+(3-5)^2=1+9+4=14

QED

RonL - Dec 22nd 2006, 12:26 PMSoroban
Hello, Jenny!

Quote:

Show that (4,5,2) ,(1,7,3),and (2,4,5) are vertices of an equilaterial triangle.

Please teach me how to solve this question.

You really should be able to reason this out yourself, Jenny.

Three (non-collinear) points always form a triangle.

What makes an equilateral triangle different from other triangles?

. . It has three equal sides, right?

How are you going to show that?'

How can we possibly find the lengths of the sides?

. . Hmmm ... how about the Distance Formula?

Your brain should be clicking along those lines effortlessly.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

How about another example?

Show that (3,0), (7,3), (4,7), (0,4) are vertices of a square.

You might make a casual graph of the points

. . so you can__see__the square.

Is your brain already working on it?

What do you know about a square?

. . It has four equal sides.

And now you know how to show that, right?

Is that all we need?

No, four equal sides will make it a*rhombus*: $\displaystyle \diamondsuit$

We need to show that the corners are right angles.

. . (We only need to show that__one__of the corners is 90°.

. . If opposite sides are equal, it is already a parallelogram.)

How do we do that?

A right angle means that two sides are*perpendicular*, right?

. . And you*do*know about perpendicular slopes, don't you?

So check out the*slopes*of two adjacent sides.

See? . . . All that is already in your brain.

. . You just have to formulate a game plan.