Question

In ^ABC, if a =36cm and m<c = 23", then what is the length of altitude h? Round your answer to the nearest whole number.

Answer 1

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. The area \( A \) of triangle \( ABC \) can be expressed in two different ways:

1. Using the base \( AB \) and height \( h \):
\[
A = \frac{1}{2} \times AB \times h
\]

2. Using the side \( a \) opposite angle \( C \) and the sine of angle \( C \):
\[
A = \frac{1}{2} \times a \times b \times \sin(C)
\]

However, as we only have \( a = 36 \, \text{cm} \) and \( m \angle C = 23^\circ \), we can compute \( A \) using the sine function.

First, to find \( \sin(23^\circ) \):
\[
\sin(23^\circ) \approx 0.3907
\]

Now, let's use the formula for the area in terms of \( a \):
\[
A = \frac{1}{2} \times a \times \sin(C) = \frac{1}{2} \times 36 \, \text{cm} \times \sin(23^\circ)
\]
\[
A = \frac{1}{2} \times 36 \, \text{cm} \times 0.3907 \approx \frac{1}{2} \times 36 \times 0.3907 \approx 7.0314 \, \text{cm}^2
\]

Now, if you want to relate this area to the altitude \( h \), we need to know the length of base \( AB \) or assume another configuration. Since base \( AB \) is not given, we will assume base \( AB \) is also \( a = 36 \, \text{cm} \) just for the sake of solving for \( h \):

We can derive \( h \) from the area:
\[
A = \frac{1}{2} \times a \times h
\]
\[
7.0314 = \frac{1}{2} \times 36 \times h
\]
\[
7.0314 = 18 \times h
\]
\[
h = \frac{7.0314}{18} \approx 0.3906 \, \text{cm}
\]

Finally, rounding \( 0.3906 \) to the nearest whole number, we get:
\[
h \approx 0 \, \text{cm}
\]

However, if more specific information about side \( AB \) is provided or if our assumption about \( AB = a \) is incorrect, please provide clarity for appropriate calculations.