What could not be the lengths of the sides of a right triangle 9ft 12ft 15ft or 4cm 7.5cm 8.5cm or 5in 10in 15in or 1.5m 2m 2.5m?

Answer 1

5" by 10" by 15" cannot be a right angled triangle because the square of the hypotenuse (longest side) must be equal to the sum of the squares of the other two sides.

Proof:

152 = 225

52 + 102 = 25 + 100 = 125

225 is not equal to 125 so the triangle is not a right angled triangle.

All the others are right-angled triangles:

92 + 122 = 152 [81 + 144 = 225]

42 + 7.52 = 8.52 [16 + 56.25 = 72.25]

1.52 + 22 = 2.52 [2.25 + 4 = 6.25]

Answer

9ft, 12ft, 15ft

9² + 12² = 81 + 144

= 225

= 15²

Which agrees with Pythagoras' Theorem, so this is a right-angled triangle.

4cm, 7.5cm, 8.5cm

4² + 7.5² = 16 + 56.25

= 72.25

= 8.5²

Which agrees with Pythagoras' Theorem, so this is a right-angled triangle.

5in, 10in, 15in

This one doesn't obey the triangle inequality, so it isn't a triangle at all. Anything that isn't a triangle isn't a right-angled triangle, and therefore we don't need to check it against Pythagoras' Theorem. But just to be thorough:

5² + 10² = 25 + 100

= 125

15² = 225

≠ 125

Which disagrees with Pythagoras' Theorem, so this is not a right-angled triangle.

1.5m, 2m, 2.5m

1.5² + 2² = 2.25 + 4

= 6.25

= 2.5²

Which agrees with Pythagoras' Theorem, so this is a right-angled triangle.