# Is a triangle with sides of lengths 24, 70 and 74 a right triangle?

Nov 14, 2015

Yes

#### Explanation:

${24}^{2} + {70}^{2} = 576 + 4900 = 5476 = {74}^{2}$

So these side lengths satisfy Pythagoras Theorem.

An alternative way of approaching this might be as follows:

$24$, $70$ and $74$ are all divisible by $2$, so this triangle is a right angled triangle if and only if a triangle with sides $12$, $35$ and $37$ is a right angled triangle.

Notice that $35 = {6}^{2} - 1$, $37 = {6}^{2} + 1$ and $12 = 2 \cdot 6$. This looks like a pattern we could check:

${\left({a}^{2} - 1\right)}^{2} = {a}^{4} - 2 {a}^{2} + 1$

${\left({a}^{2} + 1\right)}^{2} = {a}^{4} + 2 {a}^{2} + 1$

${\left(2 a\right)}^{2} = 4 {a}^{2}$

So:

${\left({a}^{2} + 1\right)}^{2} = {a}^{4} + 2 {a}^{2} + 1 = {a}^{4} - 2 {a}^{2} + 1 + 4 {a}^{2} = {\left({a}^{2} - 1\right)}^{2} + {\left(2 a\right)}^{2}$

In our case $a = 6$, but any number $a$ would work.