Two vertices of an equilateral triangle are (a,-a) and (-a,a). Find the third.

Question

The answers are two answers. Please help.

Answer 1

There are two answers because those points given make a line, over which the third point can be reflected. 

 

Treat the side made by the points given as the base of your triangle. Cut that in half. You know that the third point lies on the line perpendicular to the base that passes through that half way point. 

 

The points given are (a, -a) and (-a, a) which means that halfway point will be at the origin because (0,0) is the midpoint of those two coordinates. The slope of the line joining (a, -a) and (-a, a) will be -1. That means that the slope of the line the third point of the triangle will be on will be 1.

 

So you should draw a picture to help guide you through this:

 

Put a dash on the x axis and call it a. Put a dash on the other side of the origin and call it -a. Make sure these dashes are the same distance away from the origin or it won't make sense. Do the same thing for the y axis. Plot a point at (a, -a) and a point at (-a, a). You should have a point in the 2nd and 4th quadrants. Draw a line connecting these points. The origin is directly in the middle. Draw a line through the origin going the other way (perpendicular to the first line). This is the line on which your third point will be. It will either be in the 1st or 3rd quadrant.

 

To find out how far along that line your point should go, remember that you're dealing with a 30 - 60 - 90 triangle if you cut it from the third vertex you're trying to find to the origin. If you know the distance apart the first two points are, you know that the third point will be on this second line you drew (1/2)√3 of that value from the origin. 

 

Find the distance between those first two points: √((2a)² + (2a)²) = √(2(4a²)) = 2a√2

 

Then the third point will be ((1/2)√3)(2a√2) = a√6 away from the origin on either side.

 

Let's say that third point is (x, y). Then we want the distance between (x, y) and (0, 0) to be a√6.

 

This means that √(x² + y²) = a√6

 

We also know that the line passing through the origin with a slope of 1 is the line y = x. So we can substitute for y in this equation: √(2x²) = a√6

 

So 2x² = 6a²

 

Then x² = 3a²

 

Which means x = a√3 or x = -a√3

 

The y coordinates of those points will be the same.

 

Sorry I can't draw a picture for you. Hopefully with the description I gave you can draw it yourself (always draw a picture!). Let me know if you're still struggling.