To find the length of side \( a \) in a right triangle where \( h = 10 \) meters is the height (or the side opposite to angle \( C \)) and \( m∠C = 39° \), we can use the sine function from trigonometry.
The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse. The formula is:
\[ \sin(C) = \frac{\text{opposite}}{\text{hypotenuse}} \]
In this case, we have:
\[ \sin(39°) = \frac{h}{a} \]
Substituting the known values:
\[ \sin(39°) = \frac{10}{a} \]
Now we can solve for \( a \):
\[ a = \frac{10}{\sin(39°)} \]
Using a calculator, we find \( \sin(39°) \):
\[ \sin(39°) \approx 0.6293 \]
Then:
\[ a \approx \frac{10}{0.6293} \approx 15.9 \]
Rounding to the nearest whole number gives us:
\[ a \approx 16 \text{ meters} \]
Therefore, the length of side \( a \) is 16 meters.
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