From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!
Why do four-legged tables sometimes wobble, but three-legged tables never wobble?
Any two points determine a line segment, which is contained on a line. Any three points determine a triangle, which can be used to define only one plane.
In the case of a table with three legs, let's consider the three points corresponding to the feet of the table. Since there's only one plane in 3-dimensional space that contains the feet of the table, the table can't wobble (unless the floor wobbles, which means that either your neighbors are rearranging furniture or there's an earthquake).
The complication comes in when you have four legs. When you add in a fourth leg, you have no guarantee that the new foot lies in the plane determined by the other three feet. If the fourth foot isn't in this plane, it's impossible for all four feet to be on the floor at the same time. In other words, your table has suddenly become a Weeble that wobbles, but doesn't fall down.
Find the missing coordinates of the triangle.
We know right off the bat that our first coordinate is the origin. That would be (0, 0). Halfway there.
We know its y coordinate has to be 0 because it's on the x-axis. Since its vertical side goes straight down from (a, b), the remaining point is the horizontal distance away from the origin. That means our final point is (a, 0).
What are the lengths of each side of the triangle? Is it an equilateral, isosceles, or scalene triangle?
The side lengths are really just distances between points, and by now, we're pros in using the distance formula. All we have to do is use it three times to find the distance between the three points. Let's plug in (7, 5) and (-4, -1) first.
One out of three. Now we'll go from (-4, -1) to (1, -6).
Not exactly equal. For now, this thing's looking scalene. One last pair of points, from (7, 5) to (1, -6).
Well, we've got side lengths of 12.53, 12.53, and 7.07. Since two of them are equal to each other, this triangle is isosceles.