Only a Genius could figure this out!?

Question

Create an equilateral triangle which has an altitude of greater than or equal to one of the sides. You can use any real equilateral triangle, hyperbolic, spherical, etc.

Answer 1

An equatiorial triangle drawn on a non-euclidean space would do the trick..ex..draw it on a convex geometry like a perfect sphere, you will have an equatorial triangle that will have equal angles BUT not 60 because in this case (non euclidean) the sum of angles will be bigger than 180 and will depend on the curvature of your geometry and now the altitude will just be of equal length as the sides of the triangle because they will all be arc circles

Answer 2

Solly, just wanted to let you know that not every "equilateral triangle is two 30/60 right triangles placed adjacent" and that "the hypotenuse of that triangle will" NOT "ALWAYS be greater than the height". Consider the following and try to open up your mind to different possibilities.

The triangle on a sphere that Ramy E suggested is perfectly good. Visualize this on a globe of the world: draw a line from the pole north to the equator and then 1/4 of the way "around the world" along the equator and then back to the pole north again. You have an equilateral triangle because all of its side obviously have equal length : 1/4 of the circumference of the sphere.

If you think about it, the altitude of the triangle (defined as the length of a segment passing through a vertex and intersecting the opposite side at a right angle) is also 1/4 of the circumference of your sphere and therefore equal to the sides.

Answer 3

The height of a triangle is NOT the length of it's side, but the distance from the base to the tip, when the line forming the base is a horizontal.

An equilateral triangle is two 30/60 right triangles placed adjacent. The hypotenuse of that triangle will ALWAYS be greater than the height.

Your triangle does not exist.

Answer 4

You have to use a Right Triangle. I believe the leg going up would be the altitude and since you said greater than OR equal to...You can use any equilateral triangle because all the sides are equal.

Am I right?

Answer 5

A right equilateral spherical triangle will indeed have an altitude equal to its side length, and an obtuse equilateral spherical triangle will have an altitude greater than its length.

Answer 6

Ah, sure it exists, just not on a plane. Draw it on a sphere.

For example, imagine a big equilateral triangle where the base runs along the Earth's equator and the two sides each go to the north pole. The altitude of this triangle from the north pole to the midpoint of the side that runs along the equator is obviously the same length as the sides.

Answer 7

How can a triangle have an altitiude? Wouldn't it then be a pyramid? An equilateral triangle has all 3 sides the same length. But that's 2 dimensional not 3??? Guess I am not a Genius :)

Answer 8

Thats physically impossible in 2-D, if you have an equalteral triangle on side would have to be smaller than the altitude, which has to be smaller then one side, in an equalateral triangle all sides are equal, so its physically impossible.

In 3-D it might work.

Answer 9

Use a triangle with all its vertices at the origin. Then all of its sides have length zero, and all of its altitudes also have length zero, which is greater than or equal to zero, as required.

Answer 10

I HAVE TWO POSSIBLE SOLUTIONS THE FIRST IS: YOU HAVE TO TRY PUTTING YOUR EQUILATERAL TRIANGLE ON IT'S SIDE, LIKE A LETTER D OR LIKE THIS SYMBOL |> . IN THIS CASE, YOUR EQUILATERAL TRIANGLE HAS A HEIGTH EQUAL TO THE LENGTH OF ONE OF THE SIDES. THE SECOND IS: IF THE LENGHT OF YOUR SIDES IS EQUAL TO ZERO, THEN THE HEIGTH IS EQUAL TO ZERO TOO.