# Vertices of equilateral triangle

• Aug 4th 2012, 04:03 AM
shiny718
Vertices of equilateral triangle
If (0,0), (a,11), (b, 37) are the vertices of an equilateral triangle, find the product of a and b.
• Aug 4th 2012, 04:15 AM
Prove It
Re: Vertices of equilateral triangle
Quote:

Originally Posted by shiny718
If (0,0), (a,11), (b, 37) are the vertices of an equilateral triangle, find the product of a and b.

Each side of the triangle is equal, so that means \displaystyle \begin{align*} \sqrt{(a - 0)^2 + (11 - 0)^2} = \sqrt{(b - a)^2 + (37 - 11)^2} = \sqrt{(b - 0)^2 + (37 - 0)^2} \end{align*}.

See if you can evaluate a and b from here.
• Aug 4th 2012, 04:25 AM
shiny718
Re: Vertices of equilateral triangle
Hi Prove it, I've tried evaluating the method u suggested above but I'm stuck with $a^2 -b^2 =1248, 2ab - b^2 =555$ and $a^2 -2ab =693$ Any tips on how to continue?
• Aug 4th 2012, 05:07 AM
Prove It
Re: Vertices of equilateral triangle
Quote:

Originally Posted by shiny718
Hi Prove it, I've tried evaluating the method u suggested above but I'm stuck with $a^2 -b^2 =1248, 2ab - b^2 =555$ and $a^2 -2ab =693$ Any tips on how to continue?

From the last equation, write b in terms of a, then substitute into another equation.
• Aug 22nd 2012, 09:26 AM
thecmd999
Re: Vertices of equilateral triangle
This method will work, but in a competition situation, you'll have to be very good at bashing. A much simpler way to do this problem is to consider (b, 37) as a rotation of 60 degrees of (a, 11). This method is much easier and requires almost no bash. You can either use trigonometric sum identities, rotations on the complex plane, or rotation mapping of matrices; they are all equivalent.
• Aug 22nd 2012, 03:32 PM
richard1234
Re: Vertices of equilateral triangle
I second thecmd999's post. For example, let

$b + 37i = re^{i \theta}$

$a + 11i = re^{i \theta + \frac{\pi}{3}$

Dividing the second equation by the first,

$\frac{a+11i}{b+37i} = e^{i \frac{\pi}{3}} = \frac{1 + \sqrt{3}i}{2}$

$(b+37i)(1 + \sqrt{3}i) = 2(a+11i)$

Expand both sides, equate real and imaginary parts. I think I recognize this problem...was it on a past AIME?
• Aug 22nd 2012, 03:34 PM
thecmd999
Re: Vertices of equilateral triangle
Yeah, this was 1994 #8.
• Aug 22nd 2012, 03:37 PM
richard1234
Re: Vertices of equilateral triangle
Ah okay. I did AIME in HS (qualified to USAMO once). So most of the past AIME questions I've seen before.