arc601 wrote:
Regarding the pythagorean triple with only a right angle and hypotenuse. It seems to me like you need at least two sides.
As you can see in the attached file it is possible to make a right angled triangle with a hypotenuse of 25 and legs that are not 15 and 20.
Appreciate, if you can read the reasoning explained above.
Moreover,
In my explanation, I tried in a different way.
Yes, you are correct you cannot assume a right angled to be triplet with only one side. But, the reasoning I mentioned is only for this problem.
I guess, you will agree AE = AD = Radius of the circle. and FD =5
For the right angled triangle AEF,
let use Pythagorean triplet in the ratio :- 15:20:25
you can see, the hypotenuse EF = 25.
Again, AE = AD radius of the circle
if AE = 20 , then AD = 20 or AD = FD + AF = 5 + 15 = 20
This fits the case .
***Remember, we won't be able to solve if the information FD =5 is not mentioned.
Another way to solve:Let x = the length of side AE
So x - 5 = the length of side AF
Applying the Pythagorean Theorem,
we get: \(x^2 + (x - 5)^2 = 25^2\)
or \(2x^2 - 10x - 600 = 0\)
Divide both sides by 2 to get: \(x^2 - 5x - 300 = 0\)
Factor: \((x - 20)(x + 15) = 0\)
Since x must be POSITIVE, so \(x = 20\)
Therefore AE = 15
Cheers!!!