Question

The lengths of the sides of a right triangle are the integers a, b and c, and these integers have no common factor. If a < b < c and (c-a):b= 5:7 then find the value of a+b+c÷3
$a + b + c \div 3$

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Answer 1

Given : The lengths of the sides of a right triangle are the integers a, b and c, and these integers have no common factor.   a < b < c and (c-a):b= 5:7  

To find :   value of a+b+c÷3

Solution:

a < b < c  

a right angle  triangle

=> c² = a² + b²

=> c² - a² =  b²

=> (c + a)(c - a) =  b²

=> (c - a)/b =  b/(c + a)

(c-a):b= 5:7  

=> (c - a)/b =  b/(c + a) = 5/7

=> 7c - 7a  = 5b      &  7b = 5c  +  5a

Multiplying 1st by 5 & 2nd by 7

=> 35c - 35a  = 25b   &  49b = 35c + 35a  

=>70c = 74b

=> 35c = 37b

integers have no common factor.

=> c = 37  ,  b = 35

also 70a = 24b

=> 35a   = 12b

=> b = 35  ,  a  = 12

a = 12  , b = 35  , c = 37

Verifying 37² = 12² + 35²  

1369 = 144 + 1225

=> 1369 = 1369

a = 12  , b = 35  , c = 37

a + b + c = 12 + 35 + 37  = 84

84/3 = 28

a+b+c÷3  = 28

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