To find the length of the altitude \( h \) from vertex \( C \) to side \( AB \) in triangle \( ABC \), we can use the following formula for the area \( A \) of the triangle:
\[ A = \frac{1}{2} \times b \times h \]
where \( b \) is the base of the triangle (the length of side \( AB \) in this case), and \( h \) is the altitude from vertex \( C \) to \( AB \).
We can also express the area of triangle \( ABC \) using side \( a \) and included angle \( C \):
\[ A = \frac{1}{2} \times a \times b \times \sin(C) \]
From the given information:
- \( a = 36 \) cm (length of side opposite angle \( A \))
- \( m∠C = 23^\circ \)
To find the altitude \( h \), we first need to determine the length of side \( b \) (which is the side opposite angle \( C \)). We can use the sine function to find this:
\[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]
However, we need to express \( h \) in terms of \( a \) and \( C \).
The altitude can also be calculated directly using the formula:
\[ h = a \cdot \sin(C) \]
Now substituting the values:
\[ h = 36 \cdot \sin(23^\circ) \]
Using a calculator to find \( \sin(23^\circ) \):
\[ \sin(23^\circ) \approx 0.3907 \]
Now calculate \( h \):
\[ h \approx 36 \cdot 0.3907 \approx 14.0652 \]
Rounding this to the nearest whole number, we find:
\[ h \approx 14 \text{ cm} \]
So the length of the altitude \( h \) is approximately 14 cm.