# Math Help - Triangles whose side lengths are consecutive integers

1. ## Triangles whose side lengths are consecutive integers

Hi guys, first time posting here.

I've got an investigation question which I'm not sure on how to approach.

The question is: Consider the triangles whose side lengths are consecutive integers k, k+1, k+2, with k greater than or equal to 1.

(a)Comment on the nature of the triangle formed when k=1.
(b)Show that the triangle formed when k=2, is obtuse.
(c)Show that only a right triangle is formed when k=3.

The closest theory I've found that applies to the question is the Heronian Triangle: Heronian Triangle -- from Wolfram MathWorld
I'm still not sure if it applies to my question or not. I'd appreciate any help which comes my way in solving the question posted above.

cem.

2. If there is anything unclear about the question, feel free to ask. But I cant say there is much to the question other than what is provided in my post above.

3. Originally Posted by cem
Hi guys, first time posting here.

I've got an investigation question which I'm not sure on how to approach.

The question is: Consider the triangles whose side lengths are consecutive integers k, k+1, k+2, with k greater than or equal to 1.

(a)Comment on the nature of the triangle formed when k=1.
(b)Show that the triangle formed when k=2, is obtuse.
(c)Show that only a right triangle is formed when k=3.

The closest theory I've found that applies to the question is the Heronian Triangle: Heronian Triangle -- from Wolfram MathWorld
I'm still not sure if it applies to my question or not. I'd appreciate any help which comes my way in solving the question posted above.

cem.
to a)
The triangle doesn't satisfy the triangle inequality:

If the lengths of the sides of a triangle are a, b and c then

$a+b>c$ . With your triangle it is: $a+b=c$ ! That means the vertices of the triangle are collinear.

to b)

A triangle with the sides a,b and c is

- acute if $a^2+b^2> c^2$

- a right triangle if $a^2+b^2= c^2$

- obtuse if $a^2+b^2< c^2$

to c):

$k^2+(k+1)^2=(k+2)^2$ . Solve for k.

4. Thanks a lot earboth. I think I can take care of the rest.