Answer:
Yes, the 3-4-5 triangle and the 1.5, 2, 2.5 triangles are both working right triangles.
Explanation:
I am assuming that the triangle we are given below is 1.5, 2, 2.5, and not necessarily a Pythagorean triple (we are asked to check if it is). We can check if this triangle meets the Pythagorean Theorem, which states that:
a^2+b^2=c^2
Where a and b are both the legs of the triangle and c is the hypotenuse (longest side). Now we can simply plug in 1.5, 2, and 2.5 and see if we get a working Pythagorean triple (if so, then the equation will be true).
(1.5)^2+2^2= 2.25 + 4= 6.25
and
(2.5)^2=6.25
This works! Giving that:
(1.5)^2+2^2=(2.5)^2 in the form of: a^2 + b^2=c^2
Now for the second part of the problem, how can we figure out if a 3,4, and 5 triangle is also a working Pythagorean triple by using our existing knowledge about the 1.5, 2, 2.5? Well, the 3, 4, and 5 triangle is also the right triangle! This is because every number in the set "1.5, 2, 2.5" is simply multiplied by 2, which is alike to scaled triangles. We can see so like this:
1.5*2=3
2*2=4
2.5*2=5
It is scaled up from the 1.5, 2, and 2.5 triangles! Now lets double check our work and see if we are correct.
3^2+4^2=9+16=25
5^2=25
Thus,
(3)^2+(4)^2=(5)^2
So, yes, the 3, 4, and 5, triangles and the 1.5, 2, 2.5, triangles are both working Pythagorean triples, we were able to prove this by scaling up and existing triple to find new ones. Have a great day!