Question

If b is the mean proportional between a and c, prove that a, c, a² + b², and b² + c² are proportional.

If b is the mean proportional between a and c, prove that a, c, a² + b², and b² + c² are in proportion.

Answer 1

∵ b is the mean proportional between a and c, then,

b² = a × c

⇒ b² = ac   ...(i)

Now a, c, a² + b² and b² + c² are in proportion

⇒ `a/c = (a^2 + b^2)/(b^2 + c^2)`

R.H.S.:

`(a^2 + b^2)/(b^2 + c^2)`

Substitute b² = ac

`(a^2 + b^2)/(b^2 + c^2)`

= `(a^2 + ac)/(ac + c^2)`

= `(a(a + c))/(c(a + c))`

= `a/c`

∴ `a/c = (a^2 + b^2)/(b^2 + c^2)`

Hence proved.