Question

The sides of some triangles are given below. Find out which ones are rightangled triangles?

i. 8,15,17 

ii. 11,12,15 

iii. 11,60,61 

iv. 1.5, 1.6, 1.7 

v. 40, 20, 30

Answer 1

i. 8² = 64, 15²  = 225, 17²  = 289 

Now, 64 + 225 = 289 

∴ 8^{​​​​​​​2} + 15^{​​​​​​​2}  = 17^{​​​​​​​2}  

The above expression is of the from (hypotenuse)^{​​​​​​​2}  = (base)^{​​​​​​​2}  + (height)^{​​​​​​​2}  

∴ The sides of lengths 8,15,17 will form a right-angled triangle.

ii. 11^{​​​​​​​2}  = 121, 12^{​​​​​​​2}  = 144, 15^{​​​​​​​2}  = 225 

But, 121 + 144 ≠ 225 

∴ 11^{​​​​​​​2}  + 12^{​​​​​​​2}  ≠ 25^{​​​​​​​2}  

∴ The above expression is not of the from (hypotenuse)^{​​​​​​​2}  = (base)^{​​​​​​​2}  + (height)^{​​​​​​​2}  

∴ The sides of lengths 11, 12, 15 will not form a right-angled triangle.

iii. 11^{​​​​​​​2}  = 121, 60^{​​​​​​​2}  = 3600, 61^{​​​​​​​2}  = 3721 

Now, 121 + 3600 = 3721 

∴ 11^{​​​​​​​2}  + 60^{​​​​​​​2}  = 61^{​​​​​​​2}  

∴ The above expression is of the from (hypotenuse)^{​​​​​​​2}  = (base)^{​​​​​​​2}  + (height)^{​​​​​​​2}  

∴ The sides of lengths 11, 60, 61 will form a right-angled triangle.

iv. 1.5^{​​​​​​​2}  = 2.25, 1.6^{​​​​​​​2}  = 2.56, 1.7^{​​​​​​​2}  = 2.89 

But, 2.25 + 2.56 ≠ 2.89 

∴ 1.5^{​​​​​​​2}  + 1.6^{​​​​​​​2}  ≠ 1.7^{​​​​​​​2}  

∴ The above expression is not of the from (hypotenuse)^{​​​​​​​2}  = (base)^{​​​​​​​2}  + (height)^{​​​​​​​2}  

∴ The sides of lengths 1.5, 1.6, 1.7 will not form a right-angled triangle

v. 40^{​​​​​​​2}  = 1600, 20^{​​​​​​​2}  = 400, 30^{​​​​​​​2}  = 900 But, 400 + 900 ≠ 1600 

∴ 20^{​​​​​​​2}  + 30^{​​​​​​​2}  ≠ 40^{​​​​​​​2}  

∴ The above expression is not of the from (hypotenuse)^{​​​​​​​2}  = (base)^{​​​​​​​2}  + (height)^{​​​​​​​2}  

∴ The sides of lengths 40, 20, 30 will not form a right-angled triangle.