1. ## Equilateral Trisngle

An equilateral triangle is one in which all three sides are of equal length. If two vertices of an equilateral triangle are (0, 4) and (0, 0), find the third vertex. How many of these triangles are possible?

Let me see if I understand this question.

1. Plot the given points.

2. The distance from (0, 0) to (0, 4) is 4 units.

3. To find the third vertex of the form (x, y), I must use the distance formula for points.

4. The second question tells me that there are more than one such triangles.

5. The distance from vertex 1 to vertex 2 to vertex 3 is the same.

Is any of this correct?

2. ## Re: Equilateral Trisngle

Originally Posted by harpazo
An equilateral triangle is one in which all three sides are of equal length. If two vertices of an equilateral triangle are (0, 4) and (0, 0), find the third vertex. How many of these triangles are possible?

Let me see if I understand this question.

1. Plot the given points.
Did you do that?

2. The distance from (0, 0) to (0, 4) is 4 units.
Correct.

3. To find the third vertex of the form (x, y), I must use the distance formula for points.
Or maybe geometry with your sketch.

4. The second question tells me that there are more than one such triangles.
No it doesn't. It asks you if there is more than one.

5. The distance from vertex 1 to vertex 2 to vertex 3 is the same.
Yes.

3. ## Re: Equilateral Trisngle

Originally Posted by harpazo
An equilateral triangle is one in which all three sides are of equal length. If two vertices of an equilateral triangle are (0, 4) and (0, 0), find the third vertex. How many of these triangles are possible? Let me see if I understand this question.

2. The distance from (0, 0) to (0, 4) is 4 units.

3. To find the third vertex of the form (x, y), I must use the distance formula for points.
Let $\mathcal{O} : (0,0)$ denote the origin and $\mathcal{P} : (0,4)$ be the other vertex.

Now the point $\mathcal{M} : (0,2)$ is the midpoint of $\overline{\mathcal{O} \mathcal{P} }$.
Moreover, the line $y=2$ is the perpendicular bisector of $\overline{\mathcal{O} \mathcal{P} }$ and it contains the third vertex.
The points where $y=2$ intersects the circle $x^2+y^2=4^2$ either can be the third vertex.

The two circles, $x^2+y^2=4^2~\&~x^2+(y-4)^2=4^2$ intersect in two points.
Either one of those points could be the third point of an equail

4. ## Re: Equilateral Trisngle

There are two such triangles, with the third vertex on opposite sides of the line through (0, 0) and (0, 4). Write the coordinates of the third vertex as (x, y). The square of its distance from (0, 0) is $\displaystyle x^2+ y^2= 16$. The square of its distance from (0, 4) is $\displaystyle x^2+ (y- 4)^2= x^2+ y^2- 8y+ 16= 16- 8y+ 16= 32- 8y= 16$
so $\displaystyle 8y= 16$ and y= 2. Then $\displaystyle x^2+ y^2= x^2+ 4= 16$ so $\displaystyle x^2= 12$ and $\displaystyle x= \pm 2\sqrt{3}$. The third vertices of the two triangles are $\displaystyle \left(2\sqrt{3}, 2\right)$ and $\displaystyle \left(-2\sqrt{3}, 2\right)$.

5. ## Re: Equilateral Trisngle

Get some graph paper.
Plotting the darn thing will make it clear.

6. ## Re: Equilateral Trisngle

Originally Posted by HallsofIvy
Write the coordinates of the third vertex as (x, y).
...as (X,Y)

7. ## Re: Equilateral Trisngle

sides relationship for the 30-60-90 special right triangle

8. ## Re: Equilateral Trisngle

6 replies including hints, questions, and complete solutions before Harpazo has even returned to the thread.

9. ## Re: Equilateral Trisngle

Originally Posted by Walagaster
6 replies including hints, questions, and complete solutions before Harpazo has even returned to the thread.
Well, he does have many posts to follow.

10. ## Re: Equilateral Trisngle

Originally Posted by Walagaster
6 replies including hints, questions, and complete solutions before Harpazo has even returned to the thread.
I would ask you to review Harpazo's many many other post carefully. Having done that, if you can still honestly be critical of giving him answers I will not do so.

11. ## Re: Equilateral Trisngle

Originally Posted by Plato
I would ask you to review Harpazo's many many other post carefully. Having done that, if you can still honestly be critical of giving him answers I will not do so.
Well, you didn't give a complete answer and I would agree with you if you said that, given the nature of his posts, what you have left undone might be challenge enough for him to finish. To tell the truth, I am somewhat conflicted how to post to this board. Looking back at the totality of this thread I can only conclude that my original post was just a waste of time. A leading conversation with the OP won't have time to develop. I should probably just be quiet and let it be.

12. ## Re: Equilateral Trisngle

Originally Posted by Cervesa
Well, he does have many posts to follow.
Yes plus I work 40 overnight hours. What do you think I do after being up all night?

13. ## Re: Equilateral Trisngle

Originally Posted by Walagaster
Well, you didn't give a complete answer and I would agree with you if you said that, given the nature of his posts, what you have left undone might be challenge enough for him to finish. To tell the truth, I am somewhat conflicted how to post to this board. Looking back at the totality of this thread I can only conclude that my original post was just a waste of time. A leading conversation with the OP won't have time to develop. I should probably just be quiet and let it be.
I work 40 overnight hours, overnight, which is not easy to do. I come home in the morning and sleep for most of the day. I reply when time allows.