To find the length of side \( a \) in the triangle, we can use the information provided, specifically the height \( h = 10 \) meters and angle \( \angle C = 39^\circ \).
Assuming that \( a \) is the side opposite angle \( C \), let's denote:
- \( h \) as the height from point \( B \) perpendicular to side \( AC \) (which means it's the altitude),
- \( c \) as the length of side \( AC \),
- \( a \) as the length of side \( BC \),
- \( b \) as the length of side \( AB \).
Using trigonometric relationships in the triangle, particularly the sine function, we can express the height in terms of side \( b \) (which is the side adjacent to angle \( C \)) and angle \( C \):
\[ h = b \cdot \sin(C) \]
Substituting in the values we have:
\[ 10 = b \cdot \sin(39^\circ) \]
We can solve for \( b \):
\[ b = \frac{10}{\sin(39^\circ)} \]
Now, letโs calculate \( \sin(39^\circ) \):
\[ \sin(39^\circ) \approx 0.6293 \]
Now plug this into the equation for \( b \):
\[ b = \frac{10}{0.6293} \approx 15.9 , \text{meters} \]
Next, we need to find side \( a \). From the geometry of the triangle, we can apply the cosine rule or relate it back to our earlier sine calculations. When height is involved, and we have the base, we can split the triangle into two right triangles, but since we have only one angle and the height we can use the cosine relationship.
Since \( a \) is opposite \( C \), we can find \( a \) using the definition of the cosine function:
In a right triangle, \( \cos(C) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
If we proceed with base \( c \):
\[ c = b \cdot \cos(39^\circ) \]
To find side \( a \) directly, we would have to relate these further by knowing another side; if distances combine purely in context to angle \( C \) from height:
Thus approximate \( a \approx h \cdot (\frac{1}{\sin(39^\circ)}) \),
And reconsider that depending on positions/hypotenuses together as equal measures.
However, commonly in many settings measuring \( h \) directly will be resultant length sides on similar formations:
To finalize and round \(a\):
Using the \( \tan(39^\circ) = \frac{h}{a} \): \[ \tan(39^\circ) \approx 0.8098, \] So, \[ a = \frac{10}{0.8098} \approx 12.35 , \text{meters}. \]
Thus, the rounded length of side \( a \) to the nearest whole number is 13 meters.
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