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 = 4:5, then find the value of (b + c-a).​

Answer 1

Let ABC is right angled triangle.

given, a < b < c

we know, hypotenuse is largest among them. so, we can assume c is hypotenuse of triangle. then, angle ABC = 90°

now from Pythagoras theorem,

c² = a² + b²

or, c² - a² = b²

or, (c - a)(c + a) = b²

or, (c - a)/b = b/(c + a) ......(1)

given, (c - a) : b = 4 : 5

or, (c - a)/b = b/(c + a) = 4/5 [from equation (1)]

case 1: (c - a)/b = 4/5

or, 5c - 5a = 4b

or, 5c = 4b + 5a .....(2)

case 2 : b/(c + a) = 4/5

or, 5b = 4c + 4a .....(3)

from equations (2) and (3),

4(4b + 5a) = 5(5b - 4a)

or, 16b + 20a = 25b - 20a

or, 40a = 9b

or, a/b = 9/40 ......(4)

again, 5c = 4b + 5a [ from equation (3) ]

= 4b + 5(9b/40)

= 4b + 9b/8

=( 32b + 9b)/8

or, 40c = 41b

or, b/c = 40/41 .....(5)

so , from equations (4) and (5),

a : b : c = 9 : 40 : 41

we see 9, 40, 41 doesn't have common factor. so, a = 9, b = 40 and c = 41

then value of (b + c - a)

= (40 + 41 - 9)

= 72 [ans]