The sides of certain triangles are given below. Determine which of them are right triangles.
(i) 9 cm, 16 cm, 18 cm
(ii) 7 cm, 27 cm, 25 cm
(iii) 1.4 cm, 4.8 cm, 5 cm
(iv) 1.6 cm, 3.8 cm, 4 cm
(v) (a - 1) cm, 2 √a cm, (a + 1) cm
In a right angled triangle
(Hypotenuse) 2 = (Base)2 + (Height)2
where hypotenuse is the longest side.
(i) L.H.S. = (Hypotenuse)2 = (18)2 = 324
R.H.S. = (Base)2 + (Height)2 = (9)2 + (16)2 = 81 + 256 = 337
⇒L.H.S. ≠ R.H.S.
∴It is not a right triangle.
(ii) L.H.S. = (Hypotenuse)2 = (27)2 = 729
R.H.S. = (Base)2 + (Height)2 = (7)2 + (25) 2 = 49 + 625 = 674
⇒ L.H.S. ≠ R.H.S.
∴It is not a right triangle.
(iii) L.H.S. = (Hypotenuse)2 = (5)2 = 25
R.H.S. = (Base)2 + (Height)2 = (1.4)2 + (4.8) 2 = 1.96 + 23.04 = 25
⇒ L.H.S. = R.H.S.
∴It is a right triangle.
(iv) L.H.S. = (Hypotenuse)2 = (4)2 = 16
R.H.S. = (Base)2 + (Height)2 = (1.6)2 + (3.8)2 = 2.56 + 14.44 = 17
⇒ L.H.S. ≠ R.H.S.
∴It is not a right triangle.
(v) L.H.S. = (Hypotenuse)2 = (a + 1)2
R.H.S. = (Base)2 + (Height)2 = (a-1)2 + (2√a)2 = a2 + 1-2a + 4a = a2 + 1 + 2a = (a + 1)2
⇒ L.H.S. = R.H.S.
∴It is a right triangle.