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).
Answers
Given: 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.
To find: The value of (b + c – a) ?
Solution:
- Now we have given that a < b < c and (c – a) : b = 4 : 5 ...........(i)
- Consider ABC as right angled triangle.
- Consider c is hypotenuse of triangle and angle ABC = 90°.
- Using Pythagoras theorem, we have:
c² = a² + b²
c² - a² = b²
(c - a)(c + a) = b²
(c - a)/b = b/(c + a) ......(ii)
- Putting (i) in (ii), we get:
(c - a)/b = b/(c + a) = 4/5
- Lets take first and third term, we have:
(c - a)/b = 4/5
5c - 5a = 4b
5c = 4b + 5a .....(iii)
- Lets take second and third term, we have:
b/(c + a) = 4/5
5b = 4c + 4a .....(iv)
- from equations iii and iv, we have:
4(4b + 5a) = 5(5b - 4a)
16b + 20a = 25b - 20a
40a = 9b
a/b = 9/40 ......(v)
- Now consider: 5c = 4b + 5a
5c = 4b + 5(9b/40)
5c = 4b + 9b/8
5c = (32b + 9b)/8
40c = 41b
b/c = 40/41 .....(vi)
- so , from equations v and vi, we get:
a : b : c = 9 : 40 : 41
(b + c - a) = (40 + 41 - 9)
(b + c - a) = 72
Answer:
So the value of (b + c - a) is 72.