# A problem I'm working on

Find all right angle triangles with integer sides with perimeter 60 units.

I'm working on this problem, any ideas?

Note by Swapnil Das
2 years, 7 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

We will try and get a Diophantine equation you can factor. Let $$x$$ and $$y$$ be the 2 shorter sides of the right angle triangle.

$x+y+\sqrt{x^2+y^2}=60$

$60-x-y=\sqrt{x^2+y^2}$

$3600+x^2+y^2-120x-120y+2xy=x^2+y^2$

$xy-60x-60y+1800=0$

$(x-60)(y-60)=1800$

We also have $$x, y < 60$$ since the perimeter must be 60. So, both brackets must be negative. Also, $$x, y > 0$$ since our triangle is non-degenerate. This implies that $$-60<x-60<0, -60<y-60<0$$. We can use this to remove certain factors of 1800.

Now we list all factors of 1800 which its complement that also satisfy the above ranges.

$1800 = (-36, -50), (-40, -45)$

From this, we get $$(x, y)$$ values of $$(24, 10)$$ and $$(20, 15)$$. Thus, the only triangles which satisfy are the 10-24-26 triangle and the 15-20-25 triangle.

Note: I didn't bother with solutions $$(10, 24)$$ and $$(15, 20)$$ since they gave the same triangles.

- 2 years, 7 months ago