In a right triangle, angle \( A \) measures \( 20^{\circ} \). The side opposite angle \( A \) is 10 centimeters long. Approximately how long is the hypotenuse of the triangle? 3.4 centimeters 10.6 centimeters 27.5 centimeters 29.2 centimeters

To find the hypotenuse of the right triangle when you know one angle and the length of the side opposite that angle, you can use the sine function. Here, we know angle \( A = 20^{\circ} \) and the opposite side length is 10 cm. The formula is: \[ \sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} \] Using this, we have: \[ \sin(20^{\circ}) = \frac{10}{\text{hypotenuse}} \] Calculating \( \sin(20^{\circ}) \) gives approximately 0.342. Rearranging the equation gives: \[ \text{hypotenuse} = \frac{10}{\sin(20^{\circ})} \approx \frac{10}{0.342} \approx 29.2 \, \text{cm} \] Thus, the hypotenuse is approximately 29.2 centimeters! Hypotenuses are the rock stars of triangle relationships! They’re not just the longest side; they also carry the weight of connecting the two other sides. If you're into solving real-world problems using triangles, you can encounter hypotenuses in construction, creating ramps, or even measuring distances in navigation, making for practical applications of geometry that can affect your daily life! Ever tried calculating something with a non-right triangle? That might just lead you into sine law territories where it's all about ratios of sides to angles! One common mistake is thinking you can directly use Pythagoras when it’s not a right triangle, or forgetting to convert your angle measures if you’re using trigonometric functions in a programming context. Always double-check those angles and side relationships for smoother sailing!