Vertices of Equilateral Triangle

mathxyz · 2024-03-21 16:28:05

Verify that the points (0, 0), (a, 0), and (a/2, sqrt{3}a/2) are the vertices of an equilateral triangle. Then show the midpoints of the three sides are the vertices of a second equilateral triangle.

Bob · 2024-03-21 20:57:20

Quite a bit of working here. You could find out the lengths of the three sides. For equilateral they should all be the same.

Then find the three midpoints.

Then do the equal length test again on those points.

Alternatively, you could use trig to get the angles.

There's a bit in Euclid that allows you to short cut the last bit by using parallel lines and similar triangles.

Bob

mathxyz · 2024-03-22 02:03:32

Bob wrote:

Quite a bit of working here. You could find out the lengths of the three sides. For equilateral they should all be the same.
Then find the three midpoints.
Then do the equal length test again on those points.
Alternatively, you could use trig to get the angles.
There's a bit in Euclid that allows you to short cut the last bit by using parallel lines and similar triangles.
Bob

Please, forget trigonometry which is in chapter 5 or 6 of the textbook.
Can you show me, when time allows, how to do this step by step without trigonometry? No rush at my end.

Bob · 2024-03-22 02:34:32

The first two points tell us the length of that side is 'a'.

So work out the length of the lines joining point 1 to point 3 and also point 2 to point 3.

If all three answers are the same, then the triangle is equilateral.

Bob

amnkb · 2024-03-22 07:31:25

FelizNYC wrote:

Verify that the points (0, 0), (a, 0), and (a/2, sqrt{3}a/2) are the vertices of an equilateral triangle. Then show the midpoints of the three sides are the vertices of a second equilateral triangle.

1st part: Equilaterals have all sides the same length
so plug pairs of pts into Distance Formula
2nd part: Use Midpoint Formula to find midpts
Use Distance Formula to show all new sides have same length

mathxyz · 2024-03-22 09:24:57

amnkb wrote:

FelizNYC wrote:
Verify that the points (0, 0), (a, 0), and (a/2, sqrt{3}a/2) are the vertices of an equilateral triangle. Then show the midpoints of the three sides are the vertices of a second equilateral triangle.
1st part: Equilaterals have all sides the same length
so plug pairs of pts into Distance Formula
2nd part: Use Midpoint Formula to find midpts
Use Distance Formula to show all new sides have same length

Can you work this one out for me as extra study notes?

Bob · 2024-03-22 21:03:52

Call the points A (0,0) B (a,0) and C (a/2, sqrt{3}a/2 )

AB = a No need to use pythag here, as the distance is straight along the x axis.

Exercise for you.

Work out BC similarly. (If you get 'a' again, you've done it.)

Bob

mathxyz · 2024-03-23 04:14:40

Bob wrote:

Call the points A (0,0) B (a,0) and C (a/2, sqrt{3}a/2 )
AB = a No need to use pythag here, as the distance is straight along the x axis.
Exercise for you.
Work out BC similarly. (If you get 'a' again, you've done it.)
Bob

Is this the set up to find BC?

(BC)^2 = [(a/2) - a)]^2 + [(sqrt{3}(a)/2]^2

Bob · 2024-03-23 05:29:05

Yes.

mathxyz · 2024-03-24 09:57:47

Bob wrote:

Yes.

Thanks.

Let me see.

(BC)^2 = [(a/2) - a)]^2 + [(sqrt{3}(a)/2]^2

(BC)^2 = (a^2/4) + (3a^2/4)

(BC)^2 = 4a^2/4

(BC)^2 = a^2

sqrt{(BC)^2} = sqrt{a^2}

BC = a

Done.

What is the purpose of this exercise than algebra practice?

Last edited by mathxyz (2024-03-24 09:58:10)

amnkb · 2024-03-24 13:54:24

FelizNYC wrote:

What is the purpose of this exercise [other] than algebra practice?

probably no other pt than to give you a chance to
(1) work some more with the fomrulas
(2) get practice seeing when formulas might be useful

mathxyz · 2024-03-28 09:50:04

amnkb wrote:

FelizNYC wrote:
What is the purpose of this exercise [other] than algebra practice?
probably no other pt than to give you a chance to
(1) work some more with the fomrulas
(2) get practice seeing when formulas might be useful

Oh, I see.

KerimF · 2024-03-29 06:23:23

The main, if not crucial, role of the human scientific brain is to find out the formula, the function or the equation which can emulate/reflect, as possible, a real problem that needs to be solved.

Therefore, the purpose of the various math's exercises is to gain, also as possible, the logical reasoning on how to get the required/needed results from analyzing a formula/function of interest or solving a well-defined equation(s). Fortunately, there are also many ready-made tools to assist someone in doing this to save time (after he learns how to use them).

mathxyz · 2024-03-29 08:34:14

KerimF wrote:

The main, if not crucial, role of the human scientific brain is to find out the formula, the function or the equation which can emulate/reflect, as possible, a real problem that needs to be solved.
Therefore, the purpose of the various math's exercises is to gain, also as possible, the logical reasoning on how to get the required/needed results from analyzing a formula/function of interest or solving a well-defined equation(s). Fortunately, there are also many ready-made tools to assist someone in doing this to save time (after he learns how to use them).

I like answering math questions. I just lack the time.

Juan · 2024-04-08 18:34:24

Let me see.

mathxyz · 2024-04-08 18:36:28

Juan wrote:

Let me see.

What?????

Bob · 2024-04-08 19:45:33

hi Juan,

My apologies if I'm making a mistake, but your posts suggest you're a robot. Please answer this question to demonstrate you're human:

What is too plus the number that comes after to.

Later edit: I've had no response so I'm going to assume Juan is a spammer and ban him. That can be reversed.

Bob

mathxyz · 2024-04-09 05:52:39

Bob wrote:

hi Juan,
My apologies if I'm making a mistake, but your posts suggest you're a robot. Please answer this question to demonstrate you're human:
What is too plus the number that comes after to ?
Bob

Block him, Bob.

amnkb · 2024-04-12 08:54:55

nycguitarguy (previous username: sologuitar) wrote:

Block him, Bob.

Isn't it nice that Bob gives people lots of chances before banning them?

mathxyz · 2024-04-12 09:23:41

amnkb wrote:

nycguitarguy (previous username: sologuitar) wrote:
Block him, Bob.
Isn't it nice that Bob gives people lots of chances before banning them?

What's your point? What does this have to do with my math problem?

Math Is Fun Forum

#1 2024-03-21 16:28:05

Vertices of Equilateral Triangle

#2 2024-03-21 20:57:20

Re: Vertices of Equilateral Triangle

#3 2024-03-22 02:03:32

Re: Vertices of Equilateral Triangle

#4 2024-03-22 02:34:32

Re: Vertices of Equilateral Triangle

#5 2024-03-22 07:31:25

Re: Vertices of Equilateral Triangle

#6 2024-03-22 09:24:57

Re: Vertices of Equilateral Triangle

#7 2024-03-22 21:03:52

Re: Vertices of Equilateral Triangle

#8 2024-03-23 04:14:40

Re: Vertices of Equilateral Triangle

#9 2024-03-23 05:29:05

Re: Vertices of Equilateral Triangle

#10 2024-03-24 09:57:47

Re: Vertices of Equilateral Triangle

#11 2024-03-24 13:54:24

Re: Vertices of Equilateral Triangle

#12 2024-03-28 09:50:04

Re: Vertices of Equilateral Triangle

#13 2024-03-29 06:23:23

Re: Vertices of Equilateral Triangle

#14 2024-03-29 08:34:14

Re: Vertices of Equilateral Triangle

#15 2024-04-08 18:34:24

Re: Vertices of Equilateral Triangle

#16 2024-04-08 18:36:28

Re: Vertices of Equilateral Triangle

#17 2024-04-08 19:45:33

Re: Vertices of Equilateral Triangle

#18 2024-04-09 05:52:39

Re: Vertices of Equilateral Triangle

#19 2024-04-12 08:54:55

Re: Vertices of Equilateral Triangle

#20 2024-04-12 09:23:41

Re: Vertices of Equilateral Triangle

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