The medium side of a \( 30- \) \( 60-90 \) right triangle is 5 cm . Find the lengths of the other two sides.
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In a \( 30-60-90 \) right triangle, the ratios of the lengths of the sides opposite the \( 30^\circ \), \( 60^\circ \), and \( 90^\circ \) angles are \( 1:\sqrt{3}:2 \). If the medium side (the side opposite the \( 60^\circ \) angle) is 5 cm, the side opposite the \( 30^\circ \) angle would be \( \frac{5}{\sqrt{3}} \approx 2.89 \) cm, and the hypotenuse (opposite the \( 90^\circ \) angle) would be \( 2 \times 5 = 10 \) cm. So, the lengths of the sides are approximately \( 2.89 \) cm for the side opposite the \( 30^\circ \) angle, 5 cm for the side opposite the \( 60^\circ \) angle, and 10 cm for the hypotenuse.
