In the given figure, \(\angle\) L = 90^{\(\circ\)}. If \(\angle\) RLM = 30^{\(\circ\)} and LR = 10 cm, find the length of LM.

Solution:

We are given a right-angled triangle \( \Delta LRM \) where \( \angle L = 90^{\circ} \). We are also given that \( \angle RLM = 30^{\circ} \) and the length of side \( LR = 10 \) cm.

We need to find the length of side \( LM \).

In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

Therefore, for \( \angle RLM = 30^{\circ} \):

\( \tan(\angle RLM) = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{LR}{LM} \)

Substitute the given values:

\( \tan(30^{\circ}) = \frac{10}{LM} \)

We know that \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \).

So, \( \frac{1}{\sqrt{3}} = \frac{10}{LM} \)

To find \( LM \), we can cross-multiply:

\( LM \cdot 1 = 10 \cdot \sqrt{3} \)

\( LM = 10\sqrt{3} \) cm.

To approximate the value:

\( \sqrt{3} \approx 1.732 \)

\( LM \approx 10 \times 1.732 = 17.32 \) cm.

Ответ: Длина LM составляет \( 10\sqrt{3} \) см (или приблизительно 17.32 см).