Question

triangle ABC has a line going through it labeled h, it also creates 2 right triangles within the regular triangle.
if h=10 meters and m angle C=39 degrees, then what is the length of side a? round your answer to the nearest whole number

Answer 1

To solve for side \( a \) in triangle ABC with the given information, we can apply trigonometry, specifically the relationship between the height and the angles of the triangle.

1. **Understand the setup**: The height \( h \) is given as \( 10 \) meters and angle \( C \) measures \( 39^\circ \).

2. **Apply trigonometric functions**: In right triangle C (where angle C is \( 39^\circ \)), the height \( h \) acts as the opposite side, and side \( a \) can be determined using the tangent function, defined as:
\[
\tan(C) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{\text{length of the adjacent side}}
\]
However, we need to express side \( a \) in terms of \( h \) and angle \( C \) through sine or cosine.

From the right triangle involving height \( h \), side \( a \) can be expressed using the sine or cosine functions:
\[
\sin(C) = \frac{h}{a}
\]

3. **Rearranging the formula**: To find side \( a \):
\[
a = \frac{h}{\sin(C)}
\]

4. **Calculate \( \sin(39^\circ) \)**:
\[
\sin(39^\circ) \approx 0.6293 \text{ (using a calculator)}
\]

5. **Substituting in values**:
\[
a = \frac{10}{\sin(39^\circ)} \approx \frac{10}{0.6293} \approx 15.9
\]

6. **Rounding to the nearest whole number**:
\[
a \approx 16
\]

Thus, the length of side \( a \) is approximately **16 meters**.