Question

In △ ABC, if a=36cm and m∠ C=23°, then what is the length of altitude? Round your answer to the nearest whole number. (1 point) 39 cm 33 cm 92 cm 14 cm

Answer 1

To find the length of the altitude from vertex A to side BC in triangle ABC, we can use the formula for the area of the triangle in terms of its sides and the sine of the included angle. The formula for the area \( A \) of a triangle is:

\[
A = \frac{1}{2}ab\sin(C)
\]

However, in this case, we only have one side \( a \) (which is opposite angle A), and the angle \( C \) (which is between sides a and b). Since we want to find the altitude from A to side BC, we can calculate the area using side \( a \) and then use the formula for the area in terms of the base and height.

Let's denote:

- \( a = 36 \, \text{cm} \)
- \( C = 23^\circ \)
- \( h \) = the altitude from A to side BC.

We can express the area of triangle ABC in two different ways:

1. Using \( a \) and the altitude \( h \):

\[
A = \frac{1}{2} \times a \times h = \frac{1}{2} \times 36 \times h
\]

2. Using two sides and the sine of the angle between them (unfortunately, since we do not have side b or angle A, we will base our calculation mainly using angle C).

Meanwhile, we can also express it using the length of side BC:

\[
A = \frac{1}{2} \times BC \times h
\]

To find BC, we need to use the sine rule or the area ratio:

1. Compute the area \( A \) using the known angle \( C \):
- First, one finds the length of side BC by setting height from A using formula.

From earlier, \( h \) = \( a \times \sin(C) \):

Thus:

\[
h = a \times \sin(C) \implies h = 36 \sin(23^\circ)
\]

Calculating this value:

\[
h = 36 \times \sin(23^\circ)
\]

The sine of 23 degrees is approximately 0.3907 (using a calculator):

\[
h \approx 36 \times 0.3907 \approx 14.0652
\]

Rounding \( h \) to the nearest whole number gives:

\[
h \approx 14 \, \text{cm}
\]

Thus, the length of altitude from vertex A to side BC is:
\[
\boxed{14 \, \text{cm}}
\]