Question

Given: △MNO, m∠M=45° m∠O=30°, MN=6 Find: NO, MO.

Answer 1

Given,

In △MNO,

⇒∠M=45°

⇒ ∠O=30

By Angle sum property,

⇒∠M + ∠N + ∠O = 180°

⇒45 + 30 + ∠N = 180°

⇒∠N = 180 - 75

⇒∠N = 105°

Let,

MN = o

MO = n

NO = m

By Sine rule,

$\frac{m}{ \sin(m) } = \frac{n}{ \sin(n) } = \frac{o}{ \sin(o) }$

Given, MN = o = 6 units.

To find : NO

$\frac{m}{ \sin(45) } = \frac{6}{ \sin(30) } \\ \\ m = 6 \times \frac{ \sin(45) }{ \sin(30 ) } \\ \\ m = 6 \times 2 \times \frac{1}{ \sqrt{2} } = 6 \sqrt{2}$

To find : MO

$\frac{n}{ \sin(n) } = \frac{o}{ \sin(o) }$

$\frac{n}{ \sin(105) } = \frac{6}{ \sin(30) } \\ \\ \frac{n}{ \sin(105) } = 12$

$n = 12 \times \sin(105) \\ \\ n = 12 \times \frac{ \sqrt{3} + 1 }{2 \sqrt{2} } \\ \\ n = 3 \sqrt{2} \times( \sqrt{3} + 1) \\ \\ n = 3 \times (\sqrt{6} + \sqrt{2} )$

Therefore,

$NO = 6 \sqrt{2} \\ MO = 3( \sqrt{6} + \sqrt{2} )$