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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UpStudy AI · Accepted Answer

1. In a $$30^\circ$$-$$60^\circ$$-$$90^\circ$$ triangle, the side lengths are in the ratio
   $$
   1 : \sqrt{3} : 2,
   $$
   where:
   - The side opposite $$30^\circ$$ is the shortest and has length $$x$$.
   - The side opposite $$60^\circ$$ (the medium side) has length $$x\sqrt{3}$$.
   - The hypotenuse (opposite $$90^\circ$$) has length $$2x$$.

2. The problem states that the medium side is $$5$$ cm. Thus, we set
   $$
   x\sqrt{3} = 5.
   $$

3. Solve for $$x$$:
   $$
   x = \frac{5}{\sqrt{3}}.
   $$
   To rationalize the denominator:
   $$
   x = \frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}.
   $$

4. Therefore:
   - The shortest side (opposite $$30^\circ$$) is
     $$
     \frac{5\sqrt{3}}{3} \text{ cm}.
     $$
   - The hypotenuse (opposite $$90^\circ$$) is
     $$
     2x = 2 \cdot \frac{5}{\sqrt{3}} = \frac{10}{\sqrt{3}} = \frac{10\sqrt{3}}{3} \text{ cm}.
     $$

The lengths of the other two sides are $$\frac{5\sqrt{3}}{3}$$ cm and $$\frac{10\sqrt{3}}{3}$$ cm.
The other two sides are $$\frac{5\sqrt{3}}{3}$$ cm and $$\frac{10\sqrt{3}}{3}$$ cm.

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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