Question

tess created a pythagorean triple of (24, 32, 40) by multiplying the known pythagorean triple 3, 4, and 5 by 8. is tess correct? can a pythagorean triple be created using multiples of a known pythagorean triple.
no, pythagorean triples only exist with small numbers.
yes, as long as each number in the pythagorean triple is multiplied by a different whole number.
no, multiplying a known pythagorean triple by a whole number will not create a pythagorean triple.
yes, as long as each number in the pythagorean triple is multiplied by the same whole number.

Explanation:

Step1: Recall Pythagorean theorem

For a Pythagorean triple \((a,b,c)\), \(a^{2}+b^{2}=c^{2}\). Given the original triple \((3,4,5)\), \(3^{2}+4^{2}=9 + 16=25=5^{2}\).

Step2: Check the new triple

For the new triple \((24,32,40)\) (where \(24 = 3\times8\), \(32=4\times8\), \(40 = 5\times8\)), \((24)^{2}+(32)^{2}=576+1024 = 1600\) and \((40)^{2}=1600\).

Step3: Generalize the property

If \((a,b,c)\) is a Pythagorean triple (\(a^{2}+b^{2}=c^{2}\)), and we multiply each number by \(k\) (a whole number), then \((ka)^{2}+(kb)^{2}=k^{2}(a^{2}+b^{2})=k^{2}c^{2}=(kc)^{2}\)

Answer:

Yes, as long as each number in the Pythagorean triple is multiplied by the same whole number.