To find the length of the altitude \( h \) from vertex \( C \) to side \( AB \) in triangle \( ABC \), we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, we can consider side \( a = BC = 36 \) cm as the base, and \( h \) as the height we want to find.
Additionally, we can express the area of the triangle using the formula involving an angle:
\[ \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \]
Here \( C = 23^\circ \), \( a = 36 \) cm and \( b \) is the length of side \( AC \). However, since we do not know \( b \), we can relate \( h \) directly to \( a \) and the sine of angle \( C \):
The area can also be expressed in terms of altitude \( h \):
\[ \text{Area} = \frac{1}{2} \times a \times h \]
Now equate both expressions for the area:
\[ \frac{1}{2} \times 36 \times h = \frac{1}{2} \times 36 \times b \times \sin(23^\circ) \]
To isolate \( h \):
\[ 36h = 36b \sin(23^\circ) \]
Simplifying gives:
\[ h = b \sin(23^\circ) \]
Finding \( b \)
To find side \( b \), we use the Law of Sines or any other method if necessary. However, since we have the information given, we can also note:
The altitude \( h \) can be calculated using:
\[ h = a \sin(C) \]
Thus substituting \( a = 36 \) cm and \( C = 23^\circ \):
\[ h = 36 \sin(23^\circ) \]
Calculating \( \sin(23^\circ) \):
\[ \sin(23^\circ) \approx 0.3907 \]
Now calculate \( h \):
\[ h \approx 36 \times 0.3907 \approx 14.1 \text{ cm} \]
Rounding to the nearest whole number gives:
\[ h \approx 14 \text{ cm} \]
Therefore, the length of the altitude \( h \) is:
14 cm
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