The sets of numbers 7, 24, 25 and 9, 40, 41 are pythagorean triples. use what you know about the pythagorean theorem and explain or show why they are pythagorean triples. be sure to show your work for each set of triples! (5 points)

Hello,
Pytagorean's triples  are like x²+y²=z² with x,y,z integers.

Let's suppose x=p²-q², y=2pq

then
x²=(p²-q²)²=p^4-2p²q²+q^4
y²=4p²q²
so
x²+y²=p^4+q^4-2p²q²+4p²q²=p^4+2p²q²+q^4=(p²+q²)²
Thus (p²-q²,2pq,p²+q²) is a Pythagorean's triple.

Examples:

p=2 and q=1 ==>p²-q²=4-1=3, 2pq=2*2*1=4,p²+q²=2²+1²=5 (3,4,5)
p=4 and q=3 ==>p²-q²=4²-3²=16-9=7, 2pq=2*4*3=24, p²+q²=4²+3²=16+9=25 (7,24,25)
...