Question

If a : b = c : d then prove it, a + c : b + d = √a²-c² : √b²-d²​

Answer 1

Answer:

$\frac{a + c}{b + d} = \frac{ \sqrt{ {a}^{2} - {c}^{2} } }{ \sqrt{ {b}^{2} - {d}^{2} } }$

Step-by-step explanation:

$\frac{a}{b} = \frac{c}{d} \\ \\ \frac{a}{c} = \frac{b}{d} = k \\ \\ \\ a = ck \\ \\ b = dk\\ applying \: componendo \: and \: dividendo \\ \\ \frac{a + c}{c} = \frac{b + d}{d} \\ \\ \frac{a + c}{b + d} = \frac{c}{d} \: \: ....(i)\\ \\ rhs \: = \: \frac{ \sqrt{ {a}^{2} - {c}^{2} } }{ \sqrt{ {b}^{2} - {d}^{2} } } \\ \\ = \frac{ \sqrt{ {(ck)}^{2} - {c}^{2} } }{ \sqrt{{(dk)}^{2} - {d}^{2}} } \\ \\ = \frac{c}{d} \times \frac{ \sqrt{ {k}^{2} - 1} }{ \sqrt{ {k}^{2} - 1 } } \\ \\ \frac{c}{d} \: \: \: ....(ii) \\ \\ from \: \: \: (i) \: \: \: \: and \: \:( ii) \\ \\ \\ \\ therefore \\ \\ \frac{a + c}{b + d} = \frac{ \sqrt{ {a}^{2} - {c}^{2} } }{ \sqrt{ {b}^{2} - {d}^{2} } } \\ \\ \\ hence \: proved....$