Question

please solve this by properties of proportion

Answer 1

Answer:

$\frac{a + c}{b + d} = \sqrt{ \frac{ {a}^{2} + {c}^{2} }{ {b}^{2} + {d}^{2} } } \\ \\ hence \: \: \: proved.$

Step-by-step explanation:

$\frac{a}{b} = \frac{c}{d} \\ \\ \frac{a}{c} = \frac{b}{d} \\ \\ \\ by \: applying \: componendo \: \: \: \\ \\ \frac{a + c}{c} = \frac{b + d}{d} \\ \\ \frac{a + c}{b + d} = \frac{c}{d} \: \: \: \: \: equation \: \: (i) \\ \\ \\ \\ \\ \\ again \: \: \: \frac{a}{b} = \frac{c}{d} \\ \\ \\ \frac{a}{c} = \frac{b}{d} \\ \\ \frac{ {a}^{2} }{ {c}^{2} } = \frac{ {b}^{2} }{ {d}^{2} } \\ \\ \\ by \: applying \: \: componendo \\ \\ \\ \frac{ {a}^{2} + {c}^{2} }{ {c}^{2} } = \frac{ {b}^{2} + {d}^{2} }{ {d}^{2} } \\ \\ \\ \frac{ {a}^{2} + {c}^{2} }{ {b}^{2} + {d}^{2} } = \frac{ {c}^{2} }{ {d}^{2} } \\ \\ \sqrt{ \frac{ {a}^{2} + {c}^{2} }{ {b}^{2} + {d}^{2} } } = \frac{c}{d} \: \: \: \: equation \: \: \: \: (ii) \\ \\ \\ equating \: \: \:( i) \: \: \: and \: \: (ii) \\ \\ \\ \\ \\ \\ \frac{a + c}{b + d} = \sqrt{ \frac{ {a}^{2} + {c}^{2} }{ {b}^{2} + {d}^{2} } } \\ \\ hence \: \: \: proved.$