MHB Infinite Isosceles Triangles w/ Integer Sides & Areas

[kaliprasad]

Show that there are infiinite isosceles triangles which have integer sides and integer areas

 

[I like Serena]

My solution:

Spoiler

Consider the isosceles triangle with legs of length $5k$, base $6k$, and height $4k$, where $k$ is any natural number.

[TIKZ]
\draw[gray] (0,0) rectangle (-0.3,0.3);
\draw[thick] (0,0) -- node[below] {3k} (3,0) -- (0,4) -- node

{5k} (-3,0) -- node[below] {3k} (0,0) -- node

{4k} (0,4);
[/TIKZ]

Its area is $\frac 12 \cdot 6k \cdot 4k = 12k^2$, which is an integer.
Therefore there are infinitely many isosceles triangles with integer sides and integer areas.

Additionally, there are infinitely many such triangles of which no two are similar to each other.
That's because there are infinitely many primitive pythagorean triples (of which the classical 3-4-5 is an example).
​

​

 

[kaliprasad]

My solution:

Spoiler

Consider the isosceles triangle with legs of length $5k$, base $6k$, and height $4k$, where $k$ is any natural number.

[TIKZ]
\draw[gray] (0,0) rectangle (-0.3,0.3);
\draw[thick] (0,0) -- node[below] {3k} (3,0) -- (0,4) -- node

{5k} (-3,0) -- node[below] {3k} (0,0) -- node

{4k} (0,4);
[/TIKZ]

Its area is $\frac 12 \cdot 6k \cdot 4k = 12k$, which is an integer.
Therefore there are infinitely many isosceles triangles with integer sides and integer areas.

Additionally, there are infinitely many such triangles of which no two are similar to each other.
That's because there are infinitely many primitive pythagoreans triples (of which the classical 3-4-5 is an example).
​

​

Spoiler

area = $12k^2$