Question

What is the length of segment AC? Round your answer to the nearest hundredth.

triangles ABC and ABD in which the triangles share segment AB and angle B is a right angle, the measure of angle CAB is 34 degrees, the measure of angle BDA is 31 degrees, and the measure of segment AB is 3 units

2.49 units
3.62 units
4.48 units
5.36 units

Answer 1

To find the length of segment AC in triangle ABC, we can use the properties of right triangles and trigonometric functions. We have the following information:

- Triangle ABC is a right triangle with angle B as the right angle.
- Angle CAB = 34 degrees.
- Segment AB = 3 units.

First, we will find the length of segment AC using the sine function.

Since angle CAB is opposite to segment BC, we can use the sine:

\[
\sin(\text{angle CAB}) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AC}
\]

We do not yet know BC, but we can express BC in terms of AC. Using the cosine function for angle CAB, we have:

\[
\cos(\text{angle CAB}) = \frac{AB}{AC}
\]

We know that AB = 3, so we can write:

\[
\cos(34^\circ) = \frac{3}{AC}
\]

Now solving for AC yields:

\[
AC = \frac{3}{\cos(34^\circ)}
\]

Calculating \(\cos(34^\circ)\) using a calculator:

\[
\cos(34^\circ) \approx 0.829
\]

Substituting this value in:

\[
AC \approx \frac{3}{0.829} \approx 3.62
\]

So the length of segment AC, rounded to the nearest hundredth, is approximately **3.62 units**.

Therefore, the correct choice is:

3.62 units.