Question

Does 24 69 74 make a right triangle

Answer 1

well, let's recall that in a right-triangle the pythagorean theorem applies, meaning that c² = a² + b².

well, let's take a look at the two smaller sides hmmm 24 and 69, now if we squared them will they be 74²?

`74^2 = 24^2 + 69^2\implies 5476=576+4761\implies {\Large \begin{array}{llll} \stackrel{\bigotimes}{5476\\e 5337} \end{array}}`

Answer 2

no

Explanation:

You want to know if side lengths 24, 69, 74 will make a right triangle.

Parity

One of the simplest ways to tell if three integers will make a right triangle is to look at their parity. It must always be even. That is, the sum of the numbers cannot be odd if a right triangle is to be made.

Here, the sum is 24 +69 +74 = 167, an odd number.

These lengths will not make a right triangle.

Form factor

If the sides lengths from least to greatest are a, b, c, then a form factor f can be calculated as ...

f = a² +b² -c²

f = 24² +69² -74² = 576 +4761 -5476 = -139

The interpretation of this form factor is ...

• < 0 — obtuse triangle
• = 0 — right triangle
• > 0 — acute triangle

The measure of the largest angle in this triangle is ...

θ = arccos(f/(2ab))

θ = arccos(-139/(2·24·69)) ≈ 92.405° . . . . an obtuse angle

__

Additional comment

When the form factor is zero, the side lengths satisfy the Pythagorean theorem, a necessary and sufficient condition for the triangle to be a right triangle.

About the parity:

The parity of a square number is the same as the parity of the number being squared. For the Pythagorean theorem to be satisfied, the left and right sides of the equation ...

a² +b² = c²

must have the same parity. That means the sum (a² +b² +c²) will be the sum of two numbers with the same parity, so will have even parity. In turn, this means the sum (a+b+c) must have even parity.