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

Answer 1

The value of (a+b+c)/3 is 28

Step-by-step explanation:

Given a, b, c integers are lengths of sides of right angled triangle

And, a < b < c

We know that in a right angled triangle, hypotenuse is the greatest side

Thus, by pythagoras theorem

$c^2=a2+b^2$

$\implies c^2-a^2=b^2$

$\implies (c-a)(c+a)=b^2$   ............. (1)

But given that

$\frac{c-a}{b}=\frac{5}{7}$

or, $c-a=\frac{5}{7}b$    .................... (2)

Therefore,  from (1)

$\frac{5}{7}b(c+a)=b^2$

$\implies c+a=\frac{7}{5}b$   .....................(3)

Adding equations (2) and (3)

$2c=\frac{5b}{7}+\frac{7b}{5}$

$\implies 2c=\frac{74b}{35}$

$\implies c=\frac{37b}{35}$

$\implies \frac{c}{b}=\frac{37}{35}$

Since there are no common factors between c and b we can safely say that

$c=37, b=35$

Again from (3)

$\frac{37b}{35}+a=\frac{7b}{5}$

$\implies a=\frac{7b}{5}-\frac{37b}{35}$

$\implies a=\frac{12b}{35}$

$\implies \frac{a}{b}=\frac{12}{35}$

Since there are no common factors between a and b

Therefore,

$a=12, b=35$

Thus,

$\frac{a+b+c}{3}=\frac{12+35+37}{3}=\frac{84}{3}=28$

Hope this answer is helpful.

Know More:

Q: 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).

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