Right angled triangles

Geometry Level 2

Is it possible to construct a right-angled triangle where all three sides are odd integers?

No Yes

1 solution

Richard Standing
Feb 2, 2019

From Pythagoras:

$a^{2}= b^{2} + c^{2}$

Where a is the length of the hypoteneuse and b and c are the lengths of the shorter sides.

Rearranging:

$c^{2} = a^{2}-b^{2}$

$c^{2} = (a + b)(a - b)$

If $(a - b)$ is odd, then either $a$ or $b$ must be even since an odd number minus an odd number gives an even number but an odd number minus an even number, or an even number minus an odd number both give odd numbers.

If $(a - b)$ is even then $c^{2}$ and hence $c$ are even irrespectibe of the value of $(a+b)$

Therefore there is no possible arrangement of integer values such that $a$ , $b$ and $c$ are all odd.

Alternatively, if $b$ and $c$ are odd, then $b^2$ and $c^2$ are odd, so $a^2 = b^2 + c^2$ is even. Then $a$ must be even, so not all of $a$ , $b$ , $c$ can be odd.

Jon Haussmann - 2 years, 4 months ago

Right angled triangles

1 solution

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