Given a right triangle where one angle measures 60 degrees and the adjacent side to this angle is 12 units long, find the length of the hypotenuse.

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UpStudy AI · Accepted Answer

In a right triangle with one angle measuring 60 degrees, we can determine the length of the hypotenuse using trigonometric relationships.

**Given:**
- One angle is 60 degrees.
- The side adjacent to the 60-degree angle is 12 units long.

**Approach:**
1. **Trigonometric Relationship:**
   - The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse.
   - Mathematically, $$\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$.

2. **Apply to the 60-Degree Angle:**
   - $$\cos(60^\circ) = \frac{12}{\text{Hypotenuse}}$$.
   - We know that $$\cos(60^\circ) = 0.5$$.

3. **Solve for the Hypotenuse:**
   $$
   0.5 = \frac{12}{\text{Hypotenuse}} \
   \text{Hypotenuse} = \frac{12}{0.5} = 24 \text{ units}
   $$

**Conclusion:**
The length of the hypotenuse is **24 units**.

**Verification Using 30-60-90 Triangle Ratios:**
In a 30-60-90 triangle, the sides are in the ratio $$1 : \sqrt{3} : 2$$, where:
- The side opposite the 30-degree angle is the shortest ($$1x$$).
- The side opposite the 60-degree angle is $$\sqrt{3}x$$.
- The hypotenuse is $$2x$$.

Given that the side adjacent to the 60-degree angle (which corresponds to the side opposite the 30-degree angle) is 12 units:
- $$1x = 12$$ units
- Therefore, the hypotenuse $$2x = 24$$ units.

This confirms our earlier result.

**Answer:**  
The hypotenuse is 24 units long.
The hypotenuse is 24 units long.

Given a right triangle where one angle measures 60 degrees and the adjacent side to this angle is 12 units long, find the length of the hypotenuse.

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