Two vertices of an equilateral triangle are (-2,0) and (2,0). Find the third vertex.

Expert Answers
injdott eNotes educator| Certified Educator

Since the y-coordinates of both the vertices are zero, this tells you that one side of the triangle sits on the x-axis. The difference in the x-coordinates thus gives you the length, which is 4. Since the triangle is equilateral, all the other sides are also 4 in length. Also for an equilateral triangle, the height (or altitude) bisects the side it joins or intersects. The midpoint of the given side is at (0,0) which also happens to be where the y-axis crosses the x-axis at a right angle. So the height of the triangle can be drawn to sit along the y-axis, which means your third vertex sits on the y-axis. So you just need to calculate the height.

Now, the height or y-axis splits the triangle into two identical right-angled triangles. Each of these two triangles has a hypotenuse of length 4 (the other two sides of the equilateral triangle), a side of length 2 (one half the length of the given side sitting on the x-axis), and a common height or altitude. Since this is a right-angled triangle where you know the length of two of the sides you can use the Pythagorean theorem where c^2=a^2+b^2, c being the length of the hypotenuse. Re-arrange the formula to get b=squareroot of (c^2-a^2), where c=4 and a=2. The height b is then square root of 12. So the third vertex sits on the y-axis at point (0, sqrt(12)). The question doesn't specify which way the triangle points so the thrid point could  be below the x-axis which means (0, -sqrt(12)) is also a valid answer.

justaguide eNotes educator| Certified Educator

An equilateral triangle has three equal sides and the angles of the triangle are equal to `180/3 = 60` degrees. Two vertices of the equilateral triangle have been given as (-2, 0) and (2, 0). These points lie on the x-axis and their mid-point is (0,0). The distance between the two points is 4. If a perpendicular bisector is drawn to the side joining the two vertices, the length l of the bisector can be determined using the right triangle formed that has a hypotenuse of 4 and the angle opposite the bisector is 60 degrees.

sin 60 = l/4

=> l = `4*sqrt 3/2 = 2*sqrt 3`

This gives the y-coordinate of the third vertex as `2*sqrt 3` and the x-coordinate is 0. There are two triangles which satisfy the conditions in the problem. The vertex of the other triangle is `(0, -2*sqrt 3)`

The third vertex can be either `(0, 2*sqrt 3)` or `(0, -2*sqrt 3)`