Question

In ∆LMN, L = 60°, M = 30°, LM = 15 cm. Find LN and Mn​

Answer 1

Given:

In ∆LMN, L = 60°, M = 30°, LM = 15 cm.

To find:

LN and MN

Solution:

In Δ LMN, we have

∠L + ∠M + ∠N = 180° ↔ [Angle sum property]

⇒ 60° + 30° + ∠N = 180°

⇒ ∠N = 180° - 90°

⇒ ∠N = 90°

Now, by using the trigonometric ratios of a right-angled triangle, we will find the values of LN and MN,

Finding the value of LN:

Here,

θ = 30°

Perpendicular = LN

Hypotenuse = LM

∴ $sin\:30\° = \frac{perpendicular}{hypotenuse}$

$\implies sin\:30\° = \frac{LN}{LM}$

$\implies \frac{1}{2} = \frac{LN}{15}$

$\implies LN = \frac{15}{2}$

$\implies \boxed{\bold{LN = 7.5\:cm}}$

Finding the value of MN:

Here,

θ = 60°

Perpendicular = MN

Hypotenuse = LM

∴ $sin\:60\° = \frac{perpendicular}{hypotenuse}$

$\implies sin\:60\° = \frac{MN}{LM}$

$\implies \frac{\sqrt{3} }{2} = \frac{MN}{15}$

$\implies MN = \frac{15\sqrt{3} }{2}$

$\implies \boxed{\bold{MN = 7.5\sqrt{3} \:cm}}$

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