Jeremy loves right triangles that have side length 1. In fact, he loves to combine them so that the hypotenuse of one becomes the leg of the next. He does this over and over to create beautiful pictures like this:

Question

Jeremy loves right triangles that have side length 1. In fact, he loves to combine them so that the hypotenuse of one becomes the leg of the next. He does this over and over to create beautiful pictures like this:

ASY FAILED

If the longer leg of the first triangle is the square root of 11, what will be the hypotenuse of the 2014th triangle?

Answer 1

To find the hypotenuse of the 2014th triangle in Jeremy's sequence, we need to understand the pattern in how the side lengths change as the triangles are combined.

Let's start by labeling the sides of the first triangle:
- Shorter leg (adjacent to the right angle) = 1 unit
- Longer leg = Square root of 11 units

When Jeremy combines two of these triangles, he places the hypotenuse of one triangle as the leg of the next triangle.

For the first combination, the hypotenuse of the first triangle becomes the longer leg of the second triangle. So, for the second triangle:
- Longer leg = Hypotenuse of the first triangle = Square root of 11 units

For the third triangle, the longer leg of the second triangle becomes the shorter leg, and the hypotenuse remains the same:
- Shorter leg = Longer leg of the second triangle = Square root of 11 units
- Longer leg = Hypotenuse of the second triangle = Square root of 11 units

From here, the pattern continues. The shorter leg of each triangle is always the same as the previous triangle's longer leg, and the longer leg stays the same.

Based on this pattern, we can see that the side lengths of the triangles repeat every two triangles.

To find the side lengths of the 2014th triangle, we can divide 2014 by 2. The remainder will tell us which triangle in the repeating pattern we are on. In this case, the remainder is 0, so we're on the "even" triangle in the pattern.

The even triangle has the same side lengths as the first triangle, so the hypotenuse of the 2014th triangle will be the same as the hypotenuse of the first triangle, which is the longer leg: The square root of 11 units.

Therefore, the hypotenuse of the 2014th triangle is the square root of 11 units.

Answer 2

Got it

thanks everyone

Answer 3

I know what diagram you are talking about, it forms a spiral of square roots for the hypotenuse.

hypotenuse of first triangle = √2
hypotenuse of 2nd triangle = √3
hypotenuse of 3rd triangle =√4
...
hypotenuse of nth triangle = √(n-1)

so hypotenuse of 2014th triangle = √2013